Abstract

Empirical studies indicate that the transition to parenthood is influenced by an individual’s peer group. To study the mechanisms creating interdependencies across individuals’ transition to parenthood and its timing, we apply an agent-based simulation model. We build a one-sex model and provide agents with three different characteristics: age, intended education, and parity. Agents endogenously form their network based on social closeness. Network members may then influence the agents’ transition to higher parity levels. Our numerical simulations indicate that accounting for social interactions can explain the shift of first-birth probabilities in Austria during the period 1984 to 2004. Moreover, we apply our model to forecast age-specific fertility rates up to 2016.

Introduction

Human behavior, including childbearing behavior, is performed by socialized actors deeply rooted in a web of social relationships, such as those created by kinship, love, power, friendship, competition, and interest. Beliefs, norms, services, and goods are exchanged, traded, negotiated, and enforced within informal social networks of personal communities (Mitchell 1974). Within their social circle of relationships, individuals may exchange information about possibilities and consequences of specific childbearing choices, learn about other people’s preferences, form expectations on their choices, feel induced to conform to other people’s norms about family-related behavior, and modify their interpretation of a specific behavior.

Empirical evidence increasingly suggests social interaction to be an important determinant of demographic behavior. Diffusion processes are currently an integral part of the literature on fertility decline (Cleland and Wilson 1987; Knodel and van de Walle 1979; Mason 1992; Palloni 1998; Pollak and Watkins 1993; Watkins 1987).

In sociodemographic research, social determinants due to social interaction gained relevance when the empirical evidence provided by the European demographic history of the past century showed that regional patterns of fertility decline conformed very closely to linguistic, ethnic, and religious territorial boundaries. Some sociodemographers interpreted these patterns as the result of an ongoing ideational change, diffusing ideals about smaller family size across political borders but following cultural lines (Bongaarts and Watkins 1996; Watkins 1986).

Consequently, the way in which attitudes, values, and norms spread within a population became central in research on family and fertility. The effects of social interaction mechanisms have been explored by using formal micro-analytical models, whose fit with observed fertility trends confirm the potential explanatory power of social interaction mechanisms (Kohler 2000, 2001; Montgomery and Casterline 1996; Rosero-Bixby and Casterline 1993).

In all these applications, social interaction enters fertility explanations, both at the micro and macro level. Individual and population fertility are interdependent because the aggregation of individual fertility behavior produces externalities (such as the erosion of norms, pressure to conform, and path dependency of the information exchange). Kohler (2001) efficiently summarized the features of this micro-macro link: (a) social interaction can alter the distribution of knowledge in the population and affect reproductive decisions under uncertainty by conveying information on the consequences of low fertility or on the dynamics of social change; (b) it may establish a collective behavior among community members and initiate a fertility change when other factors would instead inhibit it; and (c) it may induce an endogenous transformation of social institutions and social norms.

The analysis of social mechanisms like social learning and social influence also plays an increasingly relevant role in demographic explanations of observed family formation patterns in contemporary Europe, such as the hypothesis formulated by Kohler et al. (2002) on the emergence of lowest-low fertility. However, the increasing inclusion of social interaction in the demographic theoretical framework faces a relatively unrealistic model of social learning and social influence mechanisms (Chattoe 2003). Not only are the social mechanisms not specified in a satisfactory way, but similar problems also exist in defining the influential relationships on childbearing decision-making.

This lack of precision seems to constitute a general problem in the development of demographic behavior theory. Specifically, there is a certain agreement that demography suffers from a poor level of precision in theoretical construction, a statistical modeling that is either not, or is insufficiently, theory-driven as well as the nonobservability or difficulty in observing important concepts and indicators involved in the theory (Burch 1996; de Bruijn 1999). This weakness is partially due to the inadequateness of the demographer’s methodological toolbox in answering relevant demographic questions. The very recent inclusion of agent-based simulations and systematic and comparative in-depth investigations offers new possibilities to develop and test cognitively valid behavioral theories and to speculate on the consequences of alternative micro-macro feedbacks to explain demographic patterns (Billari et al. 2006; Billari and Prskawetz 2003; Billari et al. 2003).

In this article, we introduce an agent-based model to study social interaction: in particular, endogenous network formation and its implication for changes over time in the transition to parenthood. Such a model allows us to test whether changes in age-specific fertility experienced in the past can be explained by social interactions. We can also use this model to project age-specific fertility rates.

In the next section, we introduce the theory and hypothesis of social interaction and fertility transitions and endogenous networks. The following sections are devoted to the implementation of the model, to the data that we use to calibrate our model, and to simulation results. The last section concludes our findings.

Social Interaction and Fertility: Theory and Hypothesis

Studies of fertility timing in developed countries have a strong explanatory role with respect to individual life course transitions. These contributions include educational, occupational, partnership, and geographical mobility histories. The postponement and increasing variability in these processes have often been associated with the observed delay in childbearing. To account for fertility preferences in general, family background variables—or, more generally, early life experiences—constitute key indicators (Axinn et al. 1994).

The fertility behavior of individuals depends not only on family background variables and life course paths but also on the behavior and characteristics of other individuals as transmitted through social networks. Several authors have emphasized the importance of social interactions for fertility choices (Bernardi 2003; Bongaarts and Watkins 1996; Montgomery and Casterline 1996). A comprehensive survey of fertility and social interactions is documented by Kohler (2001). To understand the divergence in the demographic behavior of different populations with relatively similar environmental conditions, Kohler argued for a combination of economic fertility theory (based on individual optimal and rational decision rules) and theories on social interaction (which incorporate the behavior of other members of the community/society). Another contribution that emphasizes the relevance of social interactions in the context of low fertility is by Kohler et al. (2002). They found that all lowest-low fertility countries have experienced a sharp increase of the age of first birth and argued that this observation cannot be explained by changing socioeconomic incentives alone: social interactions must have induced multiplier effects or multiple equilibria. Lyngstad and Prskawetz (2010) investigated whether siblings’ fertility decisions influence each other on the basis of Norwegian register data. Their results indicated that cross-siblings effects are relatively strong for the respondents’ first birth, but weak for the second parity transition.

So far, two avenues have been explored in the literature. On the empirical side, studies have aimed to identify or control for potential effects of social interaction on fertility (Bernardi 2003; Bernardi et al. 2007). On the more formal side, models have been developed that exogenously define the structure of the social network and investigate its implication on demographic behavior (Hernes 1972; Kohler 2001). In this article, we follow a recent development in the network literature that aims to endogenously build up the social network structure (Barabasi and Albert 1999; Watts et al. 2002; Watts and Strogatz 1998). For this purpose, we choose the framework of an agent-based model wherein the mechanisms underlying the behavior of each agent is modeled explicitly—in particular, the choice of social network, which in turn influences the fertility decision.

For an individual, the social network (the set of “relevant others”) ideally consists of people who are close. In our context, “close” refers to a distance that represents a spatial distance but might also represent a distance in terms of kinship, age, education, professional occupation, and so on. Closer individuals are more likely to be relevant others. The size and characteristics of an individual’s social network may themselves depend on the individual’s characteristics. For instance, the number of relevant others increases with age during youth and adulthood, at least up to ages that are important for processes such as getting married or having children (Micheli 2000). The literature on social networks has further shown dependencies on additional individual characteristics and conditions under which the social network changes.

There is extensive and consistent evidence on the variation of networks by marital and parental status from cross-sectional comparative studies and longitudinal studies. Marital change (getting married or divorced) seems to be the main triggering process for changes in the network: those who experienced it replaced almost all (94%) of their network (Wellman et al. 1997). The transition to parenthood seems to affect the circle of non-kin, whose members change even in the short one-year time frame after pregnancy (Ettrich and Ettrich 1995). Hammer et al. (1982) found that among mothers, the composition of the network between kin and non-kin members depends on the employment status. The number of kin within a network is higher for nonworking mothers than for working mothers, whereas the number of non-kin network members is higher for employed mothers. Higher education or professional/managerial occupation entails a larger share of the network composed of non-kin (Moore 1990:732, Table 3).

For the construction of the social network in our model, the two characteristics age and intended education determine the affiliation to a social group. The choice of these characteristics is based on the empirical findings summarized in the preceding paragraphs. Individuals who are close are more likely to belong to the same social network. Members of the social network influence the behavior of each other through interaction. In our model, we assume that as the share of mothers within the social network increases, the desire to give birth is intensified. In the next section, we introduce the agent-based model, focusing on the implementation of the endogenous social network.

Model Implementation

We set up a one-sex model that allows us to simulate the different life cycle stages of females. Although partnership plays a major role in the transition to parenthood, we refrain from including mate-search into our model because it would increase the complexity of the model and complicate the interpretation of the results.

Each individual agent has an identity number id; three characteristics; and a social network that includes friends, siblings, and the agent’s mother. The agent’s characteristics are age x, intended education e, and parity p. These three characteristics have an impact on the agent’s fertility behavior, but only age and intended education determine the affiliation to a social network. We set the lower and upper age limit of reproduction to be equal to 15 and 49 years, respectively, and the maximum age of our agents as 100 years. Although agents older than 49 cannot give birth in our model, they still may influence other agents.

Education is an influential factor for social network formation and size (cf. the previous section) and thus is an important characteristic for our model. However, because education affects an agent’s network not just on the day of graduation but will already influence it during training, we consider intended education to capture the impact of education on an agent’s fertility. We assume that individuals already know their intended education during childhood. We distinguish three stages of education: primary and lower secondary, upper secondary, and tertiary. Based on these two characteristics—age and intended education—an adult agent chooses on average members for her social network. These members influence the agent’s decision on childbearing: that is, her parity, which constitutes the third characteristic of the agent. We use six stages of parity, 0 to 5+. An individual who gives birth to a child increases her parity by one. The agent’s desire to give birth—that is, to increase parity—is weakened or intensified by the influence of the social network snw. A summary of the agent’s characteristics and parameters is shown in Table 1.

Initial Population

We initialize the simulation with N individuals and base our simulations on Austrian data, as defined in the data section. The Austrian age distribution for females constitutes the initial age distribution. We assign the age xi in a deterministic way such that the share of each one-year age group in the artificial population complies with Austrian census data. Then we randomly draw a desired education level ei from the Austrian age-specific educational distribution for females at age 30 (see the Data section), . We assume that the educational distribution at age 30 determines the intended education at earlier ages because most people finish their education before they turn age 30 and the distribution of actually achieved levels of education approximates the distribution of intended education. On the basis of the assigned age and educational level, each agent is randomly assigned her initial parity, pi, according to the empirical distribution of Austrian parities by age and education, .

For agents with parity greater than or equal to 1, an age at first birth a is assigned according to the education-specific distribution of age at first births (cf. the Data section). After all initial agents have been assigned their individual characteristics, adult agents create their social network by choosing relevant others based on the characteristics age and intended education. The average parity by age and (intended) education of the initial population is plotted in Fig. 1.

Simulation Steps

During each simulation step, each agent ages by one year and might die off with a probability according to the Austrian life table (cf. the Data section). Individuals younger than 15 are considered children. At age 15, an individual becomes an adult who builds her own social network, which includes friends chosen according to the procedure described later, and who may eventually give birth to children. We start with the empirical age and parity-specific birth probability at time t (base year), , and assume that the parenthood decision may be influenced by the agent’s peer group. This social influence is captured by the individual multiplier sii to obtain the individual probability of first birth:
formula
(1)
From various studies, we know that age of the first child is a significant factor for the second birth (Prskawetz and Zagaglia 2005). The risk of second birth is higher when the first child is still young. Therefore, for the choice of another child, we additionally include the age of an agent’s youngest child xci,
formula
(2)

The decay function g(xci) leads to higher birth probabilities for mothers with young children.

The individual changes on the micro level result in a modified probability to give birth at the macro level. Thus, the baseline probabilities at t + 1 become
formula
(3)
where is the average of the social influence values si of all agents at age x and parity p. These updated probabilities to give birth enter into Eqs. 1 and 2 for the next time step. With the given individual birth probability, an agent increases her parity by one. Because we are working with a one-sex model, only female babies are created as new agents. Hence, we refer to the Austrian sex ratio at birth srb (see the Data section) as a multiplier for the number of new agents. Then they age each simulation step until arriving at adulthood (at age 15) when they eventually may give birth. When an agent turns 50, we assume that childbearing ceases. However, agents older than 50 may still influence adults of childbearing age.

Endogenous Social Network

In our model, the agent’s social network (the network of friends, peers, and relevant others) has an impact on the individual birth probabilities (see Eqs. 1, 2, 6, and 7). As mentioned in the introduction, our model could take into consideration that links in a social network may be based on any individual characteristic, such as age, kinship, love, power, friendship, professional occupation, and geography.

For our purpose, we consider the characteristics age x and intended education e to create a social network. Based on these two characteristics, we calculate a distance
formula
(4)
for each pair of agents i and j. The parameter ε is used to adjust for the fact that the maximum age difference between two agents is much higher than the maximum possible difference in intended education. To build up the social network, an agent chooses a distance d with probability
formula
(5)
and then picks a friend uniformly among all individuals with distance d. The constant c is a normalization parameter to make sure that the probabilities of all feasible distances sum to 1, and the parameter α determines the agent’s level of homophily. If α is assigned high values, the chance of a connection between similar individuals becomes high.1 The selecting agent is also added to the network of the selected agent. Thus, we assume a mutual friendship relation, which means that the underlying network topology is represented by an undirected graph. This procedure is repeated until the desired number of peers, s, is found. This desired network size is drawn from a log-normal distribution (Dunbar and Spoors 1995: Fig. 1) with mean and rounded to nearest integer.

The preceding algorithm is inspired by Watts et al. (2002), who introduced a searchable network taking into account that individuals partition the social world in more than one way. They applied this network to explain the process of delivering messages to a target person. While in the process of delivering a message, individuals consider anybody they know (i.e., all acquaintances with whom they interact due to any similarity, common interests, or loose contact), but they may be more selective with respect to peers with whom they discuss their childbearing decisions. Therefore, unlike Watts et al. (2002) who considered each characteristic independently, we use the combined measure Eq. 4.

Whereas the social network snw of agents in the initial population consists only of members chosen by way of the preceding algorithm, the social network of agents created during the simulation also contains each agent’s mother and siblings. Between birth and her 15th birthday, an agent does not actively select her own social network but may eventually be selected into another agent’s social network.

Social Influence and Parity Transition

The agent’s decision to change her parity status is influenced by her social network (Bernardi 2003; Bernardi et al. 2007). The propensity to have a first child increases with the share of parents within the agent’s social network. We implement the social influence process with a threshold function in the tradition of Granovetter (1978) and Schelling (1978). Another prominent branch of social influence models assumes that individuals repeatedly update their opinions as a weighted average of current opinions within their peer group (Baldassarri and Bearman 2007; Deffuant et al. 2001; Flache and Mäs 2008a,b; Friedkin 1998; Hegselmann and Krause 2002). Other than such opinion dynamics models, in our framework, the individual behavior (i.e., the event of giving birth to a child) is a random process, and other individuals observe the outcome of this random process in terms of realized fertility to adopt their birth probabilities. However, the agents can never observe the other agents’ fertility intentions or their birth probabilities directly. The individuals in our model observe the fertility behavior within their peer group and are affected by this behavior.

Thereby, individuals at different life stages may be affected differently. In general, individuals with a higher social status (e.g., higher education) are more prone to adhere to social norms or social influence because they gain from their status and want to avoid a loss of status. Low-status individuals, on the other hand, gain little from community membership and are, consequently, less inclined to comply with social pressure (Cole et al. 1998; Fent et al. 2008). Moreover, more-educated women anticipate more incentives from compliance to social norms in the context of competition for jobs, status, and marriage partners than less-educated women (Burke and Heiland 2006). Therefore, we model the intensity of the social influence differently for the different education groups.

Further, we argue on the basis of qualitative material and other literature that the choice between parenthood and nonparenthood is a different one than between having one or more children. Lyngstad and Prskawetz (2010) clarified that the influence of one’s peer fertility on the decision of having a first child is rather immediate because the presence of a young child might trigger the desire for motherhood. With the age of the peer’s child, the influence declines. Moreover, Lyngstad and Prskawetz showed that the social influence is relatively strong for first births but weaker and independent from the time of birth for the second parity transition. Whereas the transition to parenthood is a unique experience in which social learning plays an important role to resolve the uncertainty, at transitions to higher-order births, parents can draw on their own experiences, making peers less important. Therefore, we assume the social influence to decay exponentially for first births and additionally to be weaker for higher-order births.

Bernardi et al. (2007) demonstrated that women who already have children do not consult with childless peers in further fertility decisions. Although social influence from all peers is strong for the transition to parenthood—that is, from parity 0 to parity 1—childless peers lose their influence in the transition to higher parities for various reasons. First, individuals with parity greater than or equal to 1 already decided on becoming parents. The question whether to have at least one child is not under consideration anymore. Hence, becoming a parent is an irreversible transition. Second, social learning between childless women and mothers about the advantages and disadvantages of their choices is asymmetric in a context where childlessness is still not normative. Mothers are influenced in their decision for another child only by network members who are already parents themselves, but parents still have an impact on the childless’ decision about becoming parents for the first time. Considering the flow of influences, the network changes from an undirected graph to a directed graph. Nevertheless, in the context of a friendship or peer network (what we mean when talking about a “network”), it is still a mutual relationship.

The social influence exposed by the network affects the birth probability. Because the empirically observed birth probability shows the average birth probability at the macro level, we assume no deviation from the empirical birth probability if the parity distribution of an agent’s social network coincides with the parity distribution at the macro level.

Formally, the social influence si for an agent of parity p is modeled as a function of the difference between the share of mothers at parity pj   >   p within the social network, π, and in the whole population, π*. The social influence positively (or negatively) affects the age- and parity-specific birth probabilities (see the Data section). To determine the social influence si, we first define the relevant share of network members π whose parity exceeds the agent’s parity p. Agents who are not yet mothers consider the share of mothers (and the age of their youngest child) among their entire personal network snw,
formula
(6)
while agents who are already mothers consider the share of agents at higher levels of parity among the mothers within their network for their decision on progressing to higher parity levels,
formula
(7)
In the preceding equations, pj denotes the current parity of agent j who is a member of agent i’s social network snw, s is the size of the social network, is a decay function taking into consideration the age xcj of the youngest child of agent j, is the number of agents within the network snw with parity greater than p, and denotes the number of mothers within the network snw. Likewise, we compute the share π* of adult agents with parity greater than p on the aggregate level within the respective age group x:
formula
for the first birth and
formula
for higher-order births.
The difference between π* on the aggregate level and π on the individual level determines the social influence on an agent’s age and parity-specific birth probability bpr(x,p). We model social influence as an s-shaped function (Fig. 2) with amplitude ξ and maximum slope β,
formula
(8)

The parameters ξ and β give the intensity of the social influence when the share of network members of a specific parity diverges from the one on the aggregate level. Choosing ξ = 0 or β = 0 results in a social influence multiplier of 1 in any case, which means that the influence of the social network is completely ignored. The slope is adjusted according to the value of π* to ensure that the possible values for si always range from to . The parameter ξ describes how strongly the social network influences the agent and may range from 0 (no influence) to 2. As described earlier, higher-educated women are exposed to more pressure to conform to social norms than less-educated women. Therefore, we use different intensity values for the different education groups ξe. We further reduce ξ for second and higher-order births to model the weaker influence for higher-order births.

The multiplier given in Eq. (8) ensures that the birth probability bpr(x,p) of an agent i facing a value of π within her social network that is equal to π* on the aggregate level is not being distorted. If the micro-level share π differs from the macro-level share π*, the social influence is assigned a value in the range , assuming that positive and negative deviations are symmetric.

Choosing a social network of size s out of a population of size N with π*N possessing a certain characteristic (e.g., a parity above a certain threshold) implies that the share π defined in Eqs. (6) and (7) is a random number following a hypergeometric distribution. Transforming this random number π into a social multiplier si with the nonlinear function (8) implies that the mean values get distorted if the distribution is skewed, thereby possibly driving the average birth probabilities to higher or lower levels. Moreover, because the mean of the values for π of all agents at a given age and parity level is not necessarily equal to the respective value at the aggregate level π*, there exists another force that may modify the birth probabilities. A negative (positive) mean of (π – π*) for the respective age and parity combined with a positive (negative) skewness results in , implying that the birth probability for the respective age and parity increases (decreases) on average. If the mean of (π – π*) is positive (negative) and the skewness is positive (negative), we have two mechanisms working in the opposite direction, and therefore the size of depends on the strength of the two mechanisms. To summarize, the dynamics of the social influence depend on the skewness of the distribution of π and on the mean of (π – π*). The former depends mostly on the share of agents with a higher parity (within each age and parity group), while the latter depends on the choice of network partners. If those agents with higher parity are more frequently chosen as network partners, they have more influence, and fertility is more likely to increase. In mathematical terms, this implies that the mean of (π – π*) is positive. Thus, the same initial population can experience an increase or a decrease in fertility depending on whether the agents with a higher parity are more or less frequently chosen as network partners. Because we compute π* for each age group independently, there may even be an increase in some age groups and a decrease in other age groups at the same time.

Data

Age Distribution

For the initial population, we alternatively use the age distribution of Austrian females in 1981, 1991, or 2001 (Statistik Austria 2005a: Table 8.04, 2005b: Table 8.07) to assign age to agents in the initial population.

Distribution by Age and (Intended) Education

We assign the level of intended education according to the empirical education distribution at age 30. To account for different distributions for varying cohorts, we use the age-specific educational distribution (at age 30 and older) of Austrian females in 1981, 1991, or 2001 (Statistik Austria 1985: Table 13, 1994: Table 14, 2004: Table 15) to estimate a cohort-specific distribution for intended education. For younger cohorts, we use the educational distribution at age 30 taken from education forecasts (Anne Goujon/VID, personal communication, 2008).

We distinguish three stages of education, whereas the Austrian data we use as input distinguish six to eight stages. We therefore merge these groups as follows: (1) primary/lower secondary education encompasses basic schooling (up to 9 years) and lower secondary education (including apprenticeships and normally between 10 and 12 years of schooling), (2) upper secondary education encompasses the Austrian “Gymnasium” and its equivalents, such as corresponding nonacademic vocational training at a similar level, and (3) tertiary education (including postgraduate studies, the training of primary school and gymnasium teachers, art academies, and so on).

Distribution by Age, Education, and Parity

Based on the Austrian distribution by age, education, and parity of 1981, 1991, or 20012 (Statistik Austria 1989: Table 50, 1996: Table 48, 2005c: Table 47), we assign a corresponding parity for the initial agents.

Parity-Specific Birth Probability by Age

The birth probabilities we apply in our simulations derive from computations by Tomáš Sobotka on the basis of data provided by Statistik Austria. Unfortunately, these data are available only from 1984 onward. For simulations where we start with the 1981 initial population, we need to apply the birth probabilities of 1984. For the other experiments, we use the corresponding data from 1991 and 2001.

Age at First Birth by (Intended) Education

We use data on age at first birth, taking into account the mothers’ level of education from the census. Because these data are provided only for five-year age groups, we interpolate the data with piecewise cubic hermite polynomials to interpolate age at first birth by single years of age.

Sex Ratio at Birth

Because we do not include male agents in our model, we need the sex ratio at birth to calculate the number of new agents per simulation step. We again use Austrian data of the particular base year for this purpose.

Life Table

To simulate mortality,3 we use the death probability from the Austrian Life Table of 1981, 1991, or 2001 (Statistik Austria 1998: Table 2.38, 2005b: Table 4.25.1, Table 4.25.2).

Simulation Parameters and Results

In this section, we discuss the results from simulations of the agent-based model introduced in the previous sections. We set the population size equal to N = 50,000 and present the average over 25 simulation runs.

First, we test the sensitivity of the results to social network parameters , ε, and α. We run a set of simulations in which we vary two out of three parameter values and fix the third parameter. The education-specific influence parameters ξ1, ξ2, and ξ3 are held constant.

We start with an initial population distributed according to the Austrian population in 1981 and exposed to birth probabilities of 1984 and simulate it forward in time 20 years.4 To illustrate the performance of our agent-based model, we plot the sum of absolute differences between simulated and observed age-specific fertility rates in 2004,5 (see Fig. 3).

The parameter combinations that perform best are displayed in white, and those that perform worst are indicated in black. Our model performs best when setting the parameter α, which represents the homophily within an agent’s chosen personal network, between 0.2 and 0.4. Because α determines the extent to which the agents prefer peers of the same age and education group, it has an impact on the spillover of social influence between different subgroups of the population. A low level of α would encourage the flow of social influence between different groups, while a high level of α would hamper it. The ideal value for ε, which intensifies the education difference over the age difference for calculating the distance between two agents (Eq. 4), would be between 4 and 8 (Fig. 3, panel a). Low values of ε indicate a population where individuals are more inclined to be in contact with individuals of the same age, and high values of ε indicate that individuals are more prone to meet with individuals of the same education group. The model is less sensitive to the average network size as long as the value is above a certain minimum level of six network members (Fig. 3, panel b). If the average network size is too small the influence of a single peer may become too high, resulting in rather erratic dynamics.

For the following simulations, we set the parameter ε equal to 5; thus, the distance of two agents at the same age with a difference in the education by one level is equal to the distance of two agents with the same education and an age difference of 5 years. The homophily parameter α is set equal to 0.35, leading to a peer group that, on average, consists of 90% of members who have a distance of 7 or lower and one-third of agents with exactly the same characteristics.

The social influence parameters β and ξ give the slope and the intensity of the influence function (Eq. 8). The slope is adjusted automatically according to the value of π* to ensure that the possible values for si always range from to . The parameter ξ describes how strongly the social network influences the agent and may range from 0 (no influence) to 2. We use different intensity values for the different education groups because higher-educated women are known to be exposed to higher social pressure.

Adequate values for ξe have been identified through sensitivity analysis with respect to the parameter values and identifying the parameter region where the simulated and observed age-specific fertility rates are closest. In panels c–e of Fig. 3, we present the results of one of these sets of experiments, where one parameter ξe is fixed and the other two are varied. Moreover, we set α   =   0.35, ε   =   5, and . The simulation performs best when agents at education stage 1 are exposed to rather weak influence (Fig. 3, panels c and e), whereas the influence for education stage 2 (respectively 3) can be varied to a larger extent given that the influence of education stage 3 (respectively 2) is adjusted (Fig. 3, panel d).

In the following, we present total fertility rates and age-specific fertility rates obtained from our experiments. Because our focus is on the transition to parenthood, we also present developments of the mean age at first birth and the probability of first birth.

As a benchmark, Fig. 4 presents results of a simulation without social influence. Similar to the previous simulations, we start with an initial population distributed according to the population in 1981 and exposed to the Austrian birth probabilities of 1984 and simulate the population forward in time for 20 years. Obviously, this benchmark model can neither replicate the increase (of nearly three years) in the mean age at birth nor the decrease in the total fertility rate (Fig. 4, panels c and a). The age-specific fertility as well as the first-birth probabilities (panels b and d) stay constant and do not exhibit the typical shift during this time period. By neglecting the role of social influence, we fail to replicate the fertility development that occurred between 1984 and 2004.

By contrast, Fig. 5 shows the results of simulations that can replicate the observed fertility developments during these two decades. Again, Austrian data from 1981 provide the basis for the initial population. However, now the agents are not only endowed with age- and parity-specific birth probabilities of 1984, but they are also exposed to the influence of their personal network. Model parameters are set according to the default values (α   =   0.35, ε   =   5, , ξ1   =   0.14, ξ2   =   0.85, ξ3   =   1.25). We run our simulations for 20 years and compare the results with the empirical data provided by Statistik Austria.

Plotting the observed and simulated time series of the total fertility rate tfr validates the remarkable performance of our model (Fig. 5, panel a). The black solid line representing the simulated tfr fluctuates less than the empirical tfr, displayed as a gray dotted line. This is due to the fact that we present the average over different simulation runs to eliminate erratic fluctuations, and the simulation does not capture exogenous shocks that may have influenced fertility during that period. The trend is similar, though, with both series decreasing from 1.52 to about 1.33. Age-specific fertility rates are presented in Fig. 5, panel b. The three series show the empirical fertility rates of the base year, 1984, and the target year, 2004 (gray lines), together with the simulated rates after 20 simulation years (black dashed line). The shape of the simulated curve is nearly identical to the empirical one of 2004. A similar match of the observed shift is achieved with the age-specific probability of first birth. As shown in panel d, the probability of having a first child peaks at a considerably higher age than during the base year, and the simulated mean age at first birth increases similar to the empirically observed mean age (Fig. 5, panel c).

Another indicator for the relevance of the social influence is the varying behavior of different population groups. Figure 6 plots the distribution of age at first birth after one simulation step for the three different intended education groups. In panel a, which presents the results of our first set of simulations in which we ignore social influence, all education groups show the same behavior. By contrast, in our second simulation (panel b), which takes social influence into account, there is a pronounced distinction between agents belonging to the three education groups. Agents with higher education postpone their first birth.

In the next experiment, we initialize the model with Austrian data from 1991 and run our simulation for 15 years up to 2006. We apply the same set of parameters as in the previous simulation. Results for this experiment are summarized in Fig. 7. The simulated total fertility rate (Fig. 7, panel a) again decreases during the time period and reaches 1.40 in 2006. Panel b shows again the match between empirical and simulated data concerning the shift in age-specific fertility rates. The increase of mean age at first birth is shown in panel c, and the shift of first birth probabilities can be seen in panel d.

So far, we have demonstrated that our model is capable of reproducing shifts in the timing of fertility that occurred during the last decades. The simulation takes the empirical (age- and parity-specific) birth probabilities at the initial time as an exogenous input and produces nearly the same changes in birth probabilities that can be observed in the empirical data. Next, we apply our model to project future trends of fertility and compare our projections with the age-specific fertility assumptions applied by Statistik Austria for its recent population projection (Hanika 2006). In contrast to population forecasts, which are usually based on time-series extrapolation of recent fertility trends combined with some expert knowledge, our approach has a theoretical foundation. We use a causal model to explain trends in timing of fertility rather than continuing existing trends. Sanderson (1998) argued that combining forecasts from causal models with standard forecasts results in more accurate predictions if the forecast errors of the two different approaches are not highly correlated.

To forecast the further development of fertility, we simulate 25 years starting from 1991. Figure 8, panel a shows the simulated total fertility rates from the experiment. The age-specific fertility rates in 2016 are shown in panel b and compared with the one assumed by Statistik Austria for 2016. Our model predicts a slightly stronger shift than the population projection of Statistik Austria.

Conclusions

As recently shown by various authors (Bernardi 2003; Kohler et al. 2002), social learning and social influence play an increasing role in demographic explanations of observed family formation patterns in contemporary Europe. However, the increasing inclusion of social interaction in the demographic theoretical framework coincides with a relatively unrealistic model of the mechanisms that underlie those social interactions.

We propose to apply the methodology of agent-based models (ABMs) to study the role of social interaction for explaining observed demographic patterns. Such models allow “thought experiments that explore plausible mechanisms that may underlie observed patterns” (Macy and Willer 2002:147). Different from micro or macro simulations, ABMs provide a theoretical bridge between the micro and macro level. The dynamic bottom-up approach of ABMs—to explain global patterns by simple local interactions—is particularly useful when aiming to explain trends in fertility timing and quantum over the past decades.

In this article, we have presented an ABM on the transition to parenthood focusing on the role of social interaction and providing an endogenous formation of the social network. Calibrating our model to Austrian data, we have shown that our model captures the observed changes in the timing and quantum of fertility over the past three decades to a high degree. Sensitivity tests indicated that most network characteristics (such as homophily) are of importance when simulating peer effects, and that the intensity of social pressure—whether to conform to one’s peer fertility decisions—has a major impact on the results.

We are aware that apart from social influence there are other factors having an impact on timing and quantum of births. Socioeconomic conditions, like increasing job insecurity during early adulthood and an increase in educational attainment, provide incentives for individuals to delay childbearing (Rondinelli et al. 2006). D’Addio and d’Ercole (2005) identified two sets of influencing factors contributing to current fertility trends: (1) higher education and employment of women, and changes in patterns of family formation; and (2) shifting values of younger women toward a less traditional role of women within family and society.

Although there is clearly evidence for postponement of first births, we do not know yet to what extent recuperation may compensate for low fertility at younger ages. Qualitative research has shown that individuals perceive an influence of their peer group on their fertility intentions (Bernardi 2003; Bernardi et al. 2007). Although it is not clearly evident to what extent this influence may indeed explain realized fertility, our simulation has shown that by solely focusing on the influence of social interaction within peer groups, we can explain a decrease of fertility at younger ages and an increase of fertility at higher ages at the same time. Although contemporary population forecasts for Western European countries typically assume recuperation of fertility exogenously, in our model the same endogenous mechanism that accounts for the decrease at younger ages also explains the increase at higher ages. The direction of fertility dynamics depends on the skewness of the parity distribution and on the choice of peers (as discussed in detail earlier in the section on social influence and parity transition).

Thus, our model is able to capture the cause and the extent to which age-specific fertility changes over time. Hence, we also apply our model to forecast age-specific fertility rates. Our framework differs from common practice in population forecasts that rely on either extrapolations of past trends or expert opinions. Instead, we propose a causal model that allows us to project demographic behavior.

The next step is to apply our model to different European countries and test its validity. Within the framework of our ABM, we can experiment with alternative mechanisms that may underlie the timing and quantum of fertility in different social environments. The exploration of plausible mechanisms that underlie observed patterns is the main challenge confronting demographers as they propose efficient explanations of past trends and provide reliable projections of future demographic developments. To demonstrate the feasibility of such an approach by applying it to the topic of the transition to parenthood has been the main aim of this article.

Notes

1

Technically, this procedure is implemented such that the interval (0,1) is partitioned into subsets according to the probabilities given by Eq. 5. Then the agent draws a random number from the interval (0,1), which determines the choice of a specific value for d.

2

Analogous to the distribution by age and education, we merge the eight educational groups into three groups.

3

Mortality is not immediately relevant for the dynamics of the model. However, agents will leave the model population at some point. We apply age-specific mortality rates to model the dropping out of persons from the population rather than just removing agents at a specific age.

4

Because we have data on birth probabilities only from 1984 onward, we need to combine the 1981 census data with 1984 birth probabilities.

5

We have also investigated the performance of our model in terms of simulated first-birth probabilities and the total fertility rate, with alternative sets of education-specific influence parameters, as well as alternative goodness-of-fit measures (e.g., maximum metric, Euclidean distance). Our results are not sensitive to these variations.

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