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= Less Partnering, Less Children, or Both? |
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== Paper Overview |
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Analysis of the Drivers of First Birth Decline in Finland Since 2010 |
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image::https://i.nostr.build/1cQxSiM34VPoFT3k.png[cover, 300] |
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https://helda.helsinki.fi |
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Hellstrand, Julia Ingrid Sofia |
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2022-05 |
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Hellstrand, J I S, Nisén, J & Myrskylä, M 2022, 'Less Partnering, Less Children, or Both? Analysis of the Drivers of First Birth Decline in Finland Since 2010', European Journal of Population, vol. 38, no. 2, pp. 191-221. https://doi.org/10.1007/s10680-022-09605-8 |
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http://hdl.handle.net/10138/344264 |
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https://doi.org/10.1007/s10680-022-09605-8 |
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cc_by |
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publishedVersion |
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Downloaded from Helda, University of Helsinki institutional repository. |
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This is an electronic reprint of the original article. |
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This reprint may differ from the original in pagination and typographic detail. |
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Please cite the original version. |
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European Journal of Population (2022) 38:191–221 |
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https://doi.org/10.1007/s10680-022-09605-8 |
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NOTE: This Nostr publication is an edited version of https://core.ac.uk/download/534015729.pdf, as viewed on 11 January 2025. |
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== Less Partnering, Less Children, or Both? Analysis of the Drivers of First Birth Decline in Finland Since 2010 |
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Julia Hellstrandfootnote:Helsinki[Centre for Social Data Science and Population Research Unit, University of Helsinki, Helsinki, |
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Finland]footnote:Rostock[Max Planck Institute for Demographic Research, Rostock, Germany] · Jessica Nisénfootnote:Rostock[]footnote:[INVEST Research Flagship, University of Turku, Turku, Finland |
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] · Mikko Myrskyläfootnote:Helsinki[]footnote:Rostock[] |
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Received: 30 April 2021 / Accepted: 17 January 2022 / Published online: 14 February 2022 |
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© The Author(s) 2022 |
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mailto:julia.hellstrand@helsinki.fi[Contact Julia Hellstrand, Concerning your research paper on Finnish birthrates] |
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== Abstract |
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In the 2010s, fertility has declined in the Nordic countries, most strikingly in Finland, and first births drive the decline. It remains unclear whether this decline results from decreased fertility within unions, changing union dynamics, or both. Thus, we investigated changes in the union–first birth dynamics from 2000 through 2018 in Finland using full-coverage population register data and an incidence-based multi-state model. To do so, we calculated the yearly age-specific transition probabilities across states of single, cohabitation, marriage, and first births among 15 to 45-year-old childless men and women. We found lower fertility rates in unions after 2010, increasing dissolution rates amongst cohabiting couples, and long-term declines in the transition to marriage. Counterfactual simulations showed that, for the decline in first births since 2010, fertility within unions matters more (three-quarters) than union dynamics (one-quarter): that is, lower fertility in cohabitating and married individuals explained 42% and 13% of the decline, respectively, and decreasing fertility rates among couples entering cohabitation explained a further 17%. Decreasing marriage (19%) and cohabitation rates (2–4%) as well as higher union dissolution rates (6%) explained a smaller share of the first birth decline. The decline in first births was somewhat sharper among the lower social strata, but across strata the decreasing first birth transitions in unions explained most of the decline. To conclude, while changing union dynamics provide a partial explanation, postponing or foregoing fertility within unions represents the primary reason for the fertility decline. |
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*Keywords* First births · Union formation · Union dissolution · Finland · Incidence-based multistate model · Counterfactual approach |
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== 1 Introduction |
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In the 2010s, fertility sharply declined in many parts of Europe, particularly in the Nordic countries. The steepest of these declines occurred in Finland, where the total fertility rate (TFR) fell from 1.87 in 2010 to an all-time low of 1.35 in 2019 (Fig. 1; Official Statistics of Finland (OSF), 2019; Human Fertility Database, 2019). The Nordic fertility decline likely reflects declining lifetime fertility given that completed cohort fertility is projected to decline substantially for the first time in decades (Hellstrand et al., 2020, 2021). This projected decline is surprising, since the Nordic region previously featured a relatively high and stable cohort fertility, partly enabled by extensive social policy support provided by these countries intended to reconcile work and family life (Esping-Andersen, 2009; McDonald, 2000). Important findings identified first births as the main driver of the Nordic fertility decline (Hellstrand et al., 2020, 2021). In Finland, for instance, 75% of the decline is attributable to first births (Hellstrand et al., 2020, 2021). Therefore, a better understanding of why first births continue to decline should allow us to understand the general fertility decline. |
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Since most first births occur within unions (Jalovaara & Fasang, 2017; Kiernan, 1999), changes in union formation and union stability represent important factors in explaining fertility changes (Hiilamo, 2020). Finland and other Nordic countries serve as forerunners in shifting family formation patterns, such as lengthened single living during young adulthood, an increased progression to premarital cohabitation, and childbearing among cohabiting couples (Lesthaeghe, 2010; Surkyn & Lesthaeghe, 2004). The long-term increase in the share of the population living alone accelerated in more recent years in Finland. Specifically, in the 15to 29-year-old age group, the share living alone rose from 21.0% in 2015 to 25.9% in 2019 among men and from 18.8 to 23.9% among women Fig. 1. Simultaneously, the number of childless cohabiting couples at older reproductive ages continues to increase. However, it remains unclear how the transitions into |
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.The total fertility rate (TFR) and the share living alone by age group and gender in Finland, 1990– 2019. Source: Statistics Finland 2020, authors’ own calculations |
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image::https://i.nostr.build/QBJGPWNIL94SwV9I.png[Figure 1, 300] |
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and out of unions and from unions to first births have changed over time, and whether these changes vary based on socioeconomic status. |
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The recent fertility decline occurred after the onset of the Great Recession, yet recession indicators insufficiently explain the fertility decline (Comolli et al., 2020). The fundamental reasons for the fertility decline remain unknown, but are hypothesised to be linked to broader uncertainty beyond the actual own circumstances (Vignoli et al., 2020a, 2020b) as well as to lifestyle factors (Rotkirch, 2020). Important questions as yet unaddressed include whether and how the decline in first birth rates since 2010 relates to changes in unions. That is, first births may be decreasing due to increasing difficulties related to forming or maintaining unions, due to decreasing tendency to transition to parenthood among couples, or due to a combination of these factors. By analysing whether the decline is driven by changes in unions versus changes in fertility within unions, we can also indirectly shed light on the pertinence of the reasons hypothesised to drive the decline. Investigating trends by socioeconomic status (SES) may be helpful here, as different SES groups may change their behaviour for different reasons. Previous research shows that first birth rates have declined across all female educational groups in the Nordic countries, albeit slightly faster among the least educated (Comolli et al., 2020). Yet, the role of unions was not taken into account in these analyses. |
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This study aims to examine the extent to which the decline in first birth rates in Finland results from changes in transitions between union states (single, cohabitating, and married) and changes in first birth rates within these states. We are particularly interested in changes in union formation (the transition from single to cohabitating and from cohabitating to married), union dissolution (the transition from cohabitating or married to single), and first birth rates within unions (the transition from cohabitating or married to first birth). Our research questions are as follows: |
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1. How have union–first birth dynamics changed over time? |
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2. Is the decline in first births driven by lower fertility in unions or by changes in union patterns? |
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3. How do these changes vary by socioeconomic status? |
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To answer these questions, we estimate the age-structured transition probabilities (single, cohabiting, married, and first births) among both men and women using full-coverage Finnish population register data, and work with the probabilities within the Markov chain multistate framework. We use an incidencebased multistate model and a counterfactual approach to estimate the impact of changes in unions and first birth transitions on declining first births in Finland from 2000 through 2018. By investigating union–first birth patterns for men and women at childbearing age, our study contributes to understanding the recent fertility decline in Finland. Because Finland is often viewed as a demographic forerunner (Andersson et al., 2009), trends here might provide insights into current fertility trends more broadly. Furthermore, the Finnish population registers are exceptional even within the Nordic context in that they include detailed, longterm information on nonmarital cohabitation. |
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== 2 Background |
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2.1 Union Dynamics and First Births: Theoretical Perspectives |
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Family demographic patterns have substantially changed in high-income countries in recent decades. Since the 1960s, fertility and marriage rates have decreased from high levels, divorce rates have increased from low levels, and nonmarital cohabitation and childbearing in cohabitation have become widespread (Lesthaeghe, 2010). Furthermore, childlessness is becoming more important in shaping fertility developments in high-income countries (Kreyenfeld & Konietzka, 2017; Miettinen et al., 2015), although much of the variation across countries in total fertility currently still depends on variation in second and higher-order births (Frejka, 2008; Zeman et al., 2018). In addition to the increased availability of efficient contraception fuelling early fertility declines beginning in the 1960s (Goldin, 2006), these changes have often been attributed to changes in gender roles and shifts in attitudes and norms. More recently, broader economic uncertainty was also put forth as playing a role in fertility and marriage declines. |
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The shift from a negative to a positive relationship between female employment and fertility in the late twentieth century (Ahn & Mira, 2002; Brewster & Rindfuss, 2000) gave rise to theories predicting a U-shaped trend in fertility levels over time (Myrskylä et al., 2009, 2011; Esping-Andersen & Billari, 2015). According to these theories, improvements in gender equality would cause fertility to recover and would also strengthen the family in terms of increasing union formation and decreasing union dissolution (Goldscheider et al., 2015). As long as traditional gender roles prevail within families, the increase in women’s labour force participation is expected to increase work–family conflict among women, thereby potentially depressing fertility, increasing the risk of union dissolution, and even discouraging union formation. By contrast, when men’s involvement in the family increases, resulting in less unequal sharing of domestic chores, women’s work–family conflicts are expected to decrease, with potentially strengthening effects on families, including increased fertility (Anderson & Kohler, 2015). |
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This U-shaped prediction linking gender equality and fertility, however, is largely fuelled by cross-sectional evidence, and longitudinal analyses do not provide strong support for the prediction (Kolk, 2019). Although the gender revolution may have helped to prevent cohort fertility from decreasing to very low levels, there is little evidence that it would have increased cohort fertility (Frejka et al., 2018). One reason for the lack of a strong positive association may be that increasing gender equality could change men’s incentives for having children in ways that are poorly understood, and some of these forces may be negative. At present, cohort fertility in the Nordic countries is predicted to decline for the first time in decades, further challenging these predictions of a positive relationship (Hellstrand et al., 2020, 2021). Since there have been no signs of weakening gender equality in families or significant changes in family policies in the recent decade, the Nordic fertility decline calls for alternative explanations. |
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|
||||
The second demographic transition (SDT) theory represents another central theory to explain family changes in recent decades. While theories linking gender equality and fertility predict a reversal towards ‘more family’, SDT predicts sustained low fertility and a continuously weakening role of the family. According to SDT, changes to family formation patterns associate with shifts in attitudes and norms towards greater individual autonomy and self-actualisation (Surkyn & Lesthaeghe, 2004). The central idea is that the departure from institutional control and authority paves the way for greater individual autonomy in decision-making, whereby the emergence of ‘higher-order needs’ drives fertility decisions (Mills et al., 2011). Hence, the importance of marriage decreases, alongside the rearing of a child increasingly becoming a conscious choice taken to achieve greater personal self-fulfilment (Van De Kaa, 1987). Childbearing can be viewed as a competing event against other life goals, with women more likely postponing childbearing if they associate children with impeding their individual autonomy (Liefbroer, 2005). Changes in values are considered to result in both increased childlessness and decreases in higher parity births, given that the latter can result in further obstacles for self-fulfilment (Lesthaeghe, 2014). Additionally, greater emphasis is placed on the quality of relationships, consequently leading to the postponement of commitments and increasing tendency for unions to dissolve. According to SDT theory, shifts in demographic behaviours are first adopted by the more highly educated who possess more advanced post-materialist values, which then spread to the rest of society (Lesthaeghe & Surkyn, 1988). |
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|
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Finally, economic constraints and economic uncertainty serve as important factors explaining trends in family formation. According to microeconomic theories, couples with greater socioeconomic resources at their disposal are more likely to have (more) children given the direct costs associated with children, when such costs are not counterbalanced by higher opportunity costs (Becker, 1993). In advanced societies, individuals tend to postpone childbearing during economically uncertain times and favour it during times of economic growth (Sobotka et al., 2011). Of all parities especially first births have been affected negatively by rising economic uncertainties (Blossfeld & Hofmeister, 2006). Economic uncertainty may also have negative effects on union formation and stability (Mills & Blossfeld, 2003). |
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|
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However, the fact that the fertility decline continued in the Nordic countries during the 2010s despite macro-economic recovery has been argued to highlight the need for a broader framework of perceived uncertainty to explain fertility changes (Comolli et al., 2020). Some hypothesise that uncertainty in people’s lives stemming from globalisation dynamics, new technologies, and media channels since the onset of the Great Recession may have increased in the 2010s (Vignoli et al., 2020a, 2020b). In the narrative framework, expectations and imaginings about the future, which arise from individuals’ past experiences and shared narratives (from peers, social media, or others) and extend beyond actual economic and labour market indicators, may importantly influence fertility decision-making (Vignoli et al., 2020a, 2020b) and marriage intensions, while forming a cohabitation appears to be more compatible with such uncertainties (Guetto et al., 2020; Mills & Blossfeld, 2013). The reason why cohabitation is expected to differ from marriage and childbearing in this respect is that a cohabitation requires much less commitment than a child or a marriage, as a cohabitation can be fairly easily ended. |
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|
||||
In this study, we compare the trends in union patterns and first births in Finland within a conceptual framework built on these different lines of theory. Although this study does not aim at providing direct evidence to test these theories, it may provide some insights that are also theoretically interesting: some explanations of the fertility decline may more strongly operate through suppressing union formation or marriage, or inciting union dissolution than suppressing fertility in unions, and different SES trends may give some further hints on the mechanisms behind the fertility decline. Based on the SDT narrative, we would expect to observe a decrease in childbearing within unions and/or changing partnership dynamics (delayed unions, increasing cohabitation with a lower probability of a first birth compared to marriage, and increasing union dissolution), which are more strongly prevalent among the more highly educated. This stems from the assumption that new demographic behaviours are adopted first by the more highly educated (Lesthaeghe & Surkyn, 1988). However, although many of the changes predicted by SDT have been observed over the past five decades, currently childlessness, never partnering, union dissolution, and childbearing in cohabitation are all more strongly prevalent in lower SES groups (Jalovaara & Fasang, 2017; Jalovaara et al., 2019; Perelli-Harris et al., 2010). It can thus be debated that how relevant the SDT framework is for explaining the current fertility patterns. |
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|
||||
A decline in first births primarily explained by decreasing fertility and marriage intensities—rather than by shifting cohabitation patterns—would be in line with the uncertainty theory, as cohabitation may not be affected by uncertainty to the same extent as marriage and the entry into parenthood. There is little consensus on how perceived uncertainty can be expected to affect union dissolution (Bastianelli & Vignoli, 2021). Actual economic constraints may increase the risk of separation due to economic stress, and similarly, perceived uncertainty could also bring forth stress that strains the relationship. The reluctance to progress to a more binding relationship (i.e. marriage or parenthood) due to uncertainty could be reflected in the dissolution of cohabitation especially if the partners’ disagree on the issue. On the other hand, marriage brings predictability to the future, and hence, perceived uncertainty may not affect divorce risks. Changes in family formation patterns more strongly driven by lower SES groups may reflect obstacles to family formation due to economic constraints or uncertainty. As lower SES groups face more difficulties in finding a partner (Jalovaara & Fasang, 2017), their union formation and consequently first births may be delayed as compared to the high SES groups. They may also have a lower first birth rate because they are more likely to experience union dissolution once they have entered a union (Jalovaara, 2013). However, economic uncertainty may arise in spite of the actual own economic situation (Vignoli et al., 2020a), and consequently, a homogenous change among the SES groups is not necessarily evidence against uncertainty as an explanation for the decline. |
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|
||||
2.2 Union Dynamics and First Births: Empirical Observations in Finland |
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|
||||
Cohabitation currently represents a common first step in family formation in Finland. The vast majority of all first unions (over 90%) begin as cohabiting unions (Jalovaara, 2012), and Finland’s proportion of 20 + year-old people living in cohabiting unions is among the highest in Europe (Corselli-Nordblad & Gereoffy, 2015). Cohabiting couples exhibit much higher separation rates than married couples, but the separation rate decreases with cohabitation length, and most cohabiting couples which do not separate eventually marry (Jalovaara, 2013). When it comes to divorce, the crude divorce rate has slowly declined after reaching its peak of 2.9 in 1989 (Eurostat, 2021). |
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|
||||
First births are becoming increasingly disconnected from marriage, as first births are increasingly more likely to occur within cohabitation (Kennedy & Bumpass, 2008; Thomson & Eriksson, 2013). The share of births to unmarried women in Finland increased from 33% in 1995 to 45% in 2018, consequently placing Finland currently above the average of 42% across 28 OECD countries, but clearly lower than in other Nordic countries (OECD, 2021b). By 2018, the mean age at first birth reached 29.4 years and the mean age at first marriage 31.7 years among Finnish women (Official Statistics of Finland (OSF), 2018a, 2018b). Although first births are increasingly born to cohabiting couples, marriage and childbearing are still interconnected so that cohabiting couples who have children often marry (Rotkirch & Miettinen, 2017). A study from Iceland concludes that the order of events accounts for this change: marriage now takes place after rather than before childbearing, although marriage does not seem to be declining (Jónsson, 2020). However, married couples still have stronger childbearing intentions (Miettinen & Rotkirch, 2008) and higher first birth rates than cohabiting couples (Jalovaara & Miettinen, 2013). |
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|
||||
Compared to other high-income countries, Finland exhibits high rates of ultimate childlessness (Sobotka, 2017), and most Finns without children of their own have never lived in a stable, long-term cohabitating union (Jalovaara & Fasang, 2017; Saarela & Skirbekk, 2019). Hence, difficulties encountered in finding a partner represent one likely factor explaining high rates of childlessness in Finland. Furthermore, the contemporary gender imbalance in education, such that women are better educated than men, is particularly pronounced in Finland (OECD, 2021a). This new gender imbalance in education limits the opportunities of finding a suitable partner particularly among the least educated men who now largely outnumber women without a degree, since historically women have formed unions with men at least as well educated as themselves (Mäenpää & Jalovaara, 2015; Van Bavel, 2012). A stable partnership continues to be a strong prerequisite to childbearing also in Finland (Jalovaara & Fasang, 2017). Marriage serves as an indicator of greater union stability as opposed to cohabitation and proof of a commitment, and married men and women are much less likely to remain childless than cohabiting men and women (Saarela & Skirbekk, 2019). |
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|
||||
There are pronounced differences in union histories and the transition to parenthood between educational groups in Finland. The lower educated tend to form unions and have their first birth at younger ages than the higher educated (Jalovaara & Fasang, 2020; Nisén et al., 2014). Socioeconomic resources of both genders promote cohabitation and particularly marriage, and lower the risk of union dissolution (Jalovaara, 2012, 2013). For instance, over one third of Finns with a tertiary education had married their first cohabiting partner and were still in that marriage at age 39, while the corresponding share was 13% for those with only compulsory education (Jalovaara & Fasang, 2017). Ultimate childlessness has recently sharply increased among the less educated, and consequently, the educational gradient in childlessness among women has reversed (Jalovaara et al., 2019). Currently, both the least educated men and women are most likely to remain childless: in a late 1960s Finnish cohort consisting of individuals who completed only a basic education, over 30% of women and over 40% of men remained childless at age 40–41 (Rotkirch & Miettinen, 2017). This negative relationship between educational attainment and ultimate childlessness appears strongly driven by lower chances of the less educated to form stable unions (Nisén et al., 2018; Saarela & Skirbekk, 2019). Further, the least educated are more likely to have children in cohabitating unions while the more highly educated tend to exhibit higher first birth rates within marriage (Jalovaara & Andersson, 2018). Still, little is known about how fertility declines in the 2010s are associated with changes in unions and whether these potential changes vary based on SES. |
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|
||||
== 3 Data and Methods |
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|
||||
3.1 Data |
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|
||||
In this study, we used Finnish national longitudinal population register data compiled at Statistics Finland (permission no. TK-52-1119-17). The register data were linked to different register sources such as information on childbirths, housing and educational attainment through personal identification numbers, offering full-coverage of the entire Finnish population. The study population consists of all childless men and women aged 15–45-years-old permanently living in Finland on the last day of each year from 2000 through 2018. Individuals were followed until they had a first biological child or until they reached the age of 45. In total, the study population consists of 2 532 375 individuals and 23 847 070 person-years. Less than 0.06% of all first births were linked to two biological mothers/fathers. Consequently, the true parent for these children remained unknown. We excluded from our study 388 individuals linked to such a first birth. |
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|
||||
For each individual, data include personal information on family status (single, cohabiting, or married) at the end of each calendar year. Statistics Finland defines cohabitation as a union of two unmarried adults of the opposite sex aged 18 or older who have been living in the same dwelling for at least three months, who are not siblings or differ in age by 16 or more years (Official Statistics of Finland (OSF), 2021). An individual is considered single if s/he is not living in a cohabiting or married union. Among the study population, 2.1% of men and 1.5% of women (446 787 observations) had missing information for family status (institutionalised population and/or otherwise unclassified) and were thus excluded from the study. |
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|
||||
We formed yearly transitions for all individuals in the study population for whom personal information was available for two consecutive years. Information for two consecutive years was missing for all first entries into the study population (2 517 735 observations) and for individuals absent from the Finnish population during some period from 2000 through 2018 (130 467 observations). Furthermore, to avoid challenges related to incomplete educational data and an unknown number of unregistered first births to non-native Finns, we excluded individuals born abroad 1 (1 280 473 observations for 229 670 individuals). In total, we identified 19 468 815 yearly transitions between states (single, cohabitating, married, and first birth) for 2 125 172 individuals beginning in 2000. Among these, 740 537 were transitions to first births2, and 2 911 543 were transitions between partnership states. Appendix Table 1 provides descriptive information about first births and partnership transitions in more detail. |
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|
||||
We also estimated the transition probabilities based on SES. We considered four categories of educational attainment—primary, secondary, lower tertiary, and higher tertiary. Primary-level includes those who completed at most a lower secondary level of education (ISCED 0–2), while secondary level refers to those who completed upper secondary and post-secondary non-tertiary levels of education (ISCED 3–4). Lower tertiary includes short-cycle tertiary education and a Bachelor’s degree or the equivalent level (ISCED 5–6), while higher tertiary refers to those who completed a Master’s degree, doctoral degree or the equivalent level of education (ISCED 7–8). We used income as a complement to education as a robustness check to overcome the limitations related to using educational attainment as an explanatory variable in the period analysis—that is, currently, less educated groups include those who will later attain more advanced degrees. The income variable refers to each individual’s annual income subject to state taxation and includes both earnings and social-security benefits. Four income groups were formed based on income quartiles stratified by age, year, and gender. Because those enrolled in educational programmes in particular are known to exhibit lower birth risks (e.g. Kravdal, 1994), we performed a sensitivity analysis which excluded students (shown in the appendix). |
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|
||||
3.2 Methods |
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|
||||
We used a Markov chain multistate approach, which describes the transitions between a given set of states using transition probabilities (Briggs & Sculpher, 1998). A Markov chain evolves in discrete time and moves step-by-step from state i to state j , with the property of being memoryless. That is, the probability of each transition depends only on the state attained in the previous step and not on the history of events (Kemeny & Snell, 1971). The transition probabilities from state i to state j at a specific age and time are defined as |
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|
||||
.State transition diagram for the Markov chain |
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image::https://i.nostr.build/sFlRaApoHqolB7hS.png[Figure 2, 300] |
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|
||||
_Formula 1 as Asciimath_ |
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[stem] |
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++++ |
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p_(ij)(ag\e, t) = pr(State_t = j|State_(t−1) = i; ag\e_(t−1)) |
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++++ |
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|
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image::https://i.nostr.build/rRkC2LZrPEOjWZqJ.png[Formula 1, 300] |
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|
||||
The step size in our analyses is one year.footnote:[When we refer to a specific year, we refer to the end of that specific year.] Our state space includes the states of ‘single’, ‘cohabitating’, ‘married’, and ‘first birth. An illustration of the state space and the transitions between these states appears in Fig. 2. In our analysis, we distinguish between the transitions from ‘single’ to ‘first birth and single’ and ‘first birth and union’ in order to distinguish single parents from couples who begin cohabitating closer to the first birth event. The first birth event represents an absorbing state, meaning that once entered it cannot be left. All other states are nonabsorbing (transient) states. We estimated the yearly age-specific transition probabilities for each of the given set of states between the ages of 15 and 45 from 2000 through 2018 as |
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|
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_Formula 2 as Asciimath_ |
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[stem] |
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++++ |
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p_(ij)(x, t) = frac{"#individuals in state "j " in year " t " aged " x " and in state " i " in year "t − 1 }{"#individuals aged " x " in state " i " in year "t − 1} |
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++++ |
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image::https://i.nostr.build/CngNXtCcNthAaO62.png[Formula 2, 300] |
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|
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using simple cross tabulations.footnote:[Fitting multinomial logistic regression models to the data would be an alternative, but the results |
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would not differ.] The probabilities were estimated separately for men and women, as well as for educational and income groups, respectively. |
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|
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We used the estimated transition probabilities and counterfactual simulation[Related methodological approaches have been applied only in few prior studies: hazard ratios were |
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implemented in microsimulation models to link fertility to marital behaviours in Canada (Bélange et al., |
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2010), and a counterfactual approach was employed to examine the impact of union dissolution on fertil- |
||||
ity in Uruguay (Fernández Soto & Laplante, 2020).] to calculate what proportion of the decline in first births was attributable to changes in union dynamics versus the decline in fertility within unions.footnote:[Our counterfactual design does not necessarily take into account (changes in) the order of events. For |
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instance, if we were to observe a first birth decline attributed to changes in declining marriage rates, it |
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could in part reflect a tendency to marry increasingly after the first birth.] (For specific details, see the Technical appendix.) First, we calculated the age-specific first birth rates that would have been observed if the population in 2010 would have experienced the 2010 transition rates in the period from 2010 through 2018. We labelled this scenario ‘constant probability births’. Using the age-specific first birth rates, we calculated the proportion ever having a first birth according to a life-table approach. Second, we calculated the age-specific first birth rate and the proportion ever having a first birth that we expect to have observed if the population in 2010 would have experienced the observed changes in transition rates in the period from 2010 through 2018. We labelled this scenario ‘natural course births’. We decompose the difference between these two scenarios by changing the transition probabilities one at a time. For education groups, we adjusted the procedure to take into account that the study population progresses to higher education levels over time. Additional details appear in the Technical appendix. |
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|
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== 4 Results |
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|
||||
4.1 Age‑Specific Transition Probabilities Between States |
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||||
First, we explored how the age-specific transition probabilities have changed over time. The selected yearly age-specific transition probabilities between single, cohabitating, and married individuals, and first births for childless men and women in Finland from 2000 through 2018 appear in Fig. 3. We show the developments for two selected age groups, 25 and 35, which represent the patterns among younger and older age groups, respectively. The age-specific transition probabilities for all events appear in Appendix Fig. 7. |
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|
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4.1.1 Union Formation |
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|
||||
Since the early 2000s, the probability of forming unmarried, cohabitating unions has remained relatively stable across all ages. However, we observed a drop for the most recent years (2015–2018) among the younger ages, that is, at age 25 when the transition probability fell from 14 to 12% for men. Among women, a drop was observed only among those younger than 25. Thus, the yearly probability of remaining single has recently increased among younger individuals. The probability of marrying has exhibited a more long-term decline at nearly all ages. For instance, the yearly probability of marrying at age 25 among cohabiting women has, since the early 2000s, decreased from 11 to 6% in 2018, peaking in 2008 at 8% for age 35, and then falling to 5% in 2018. Additionally, the low probability of (directly) marrying among single individuals has decreased. |
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||||
4.1.2 Union Dissolution |
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|
||||
Married couples exhibited a lower rate of union dissolution compared to cohabiting individuals at all ages. The rate of union dissolution among married couples remains similar across all age groups, remaining relatively stable over the period of interest. Among cohabiting couples, the rate of union dissolution was higher in the younger age groups. In recent years, the probability of a union dissolving among cohabiting |
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|
||||
.Transition probabilities for single, cohabitating, and married individuals, and the first birth for childless women and men aged 25 and 35 in Finland, 2001–2018 |
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image::https://i.nostr.build/uBdypLpjQM1Hezyy.png[Figure 3, 300] |
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|
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men and women increased slightly at younger ages. That is, at age 25, the transition probability increased from 11 to 13% among women and from 14 to 15% among men from 2010 to 2018. Furthermore, the transition from marriage to cohabitation remains rare. |
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|
||||
4.1.3 Transition to First Births |
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|
||||
The age pattern in the transition to first births differs widely for cohabiting and married couples. First birth rates within cohabitating couples were highest during the early 30 s when around 10% of men and women who were cohabiting in 2017 experienced their first births in 2018. First birth rates among married men and women where highest at very young ages. Thus, first birth rates were several times higher at younger ages among married versus cohabiting couples, a difference that persisted |
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|
||||
.Transition probabilities for single, cohabitating, and married individuals, and for first births among childless women and men in 2001–2018 by level of education at ages 25 and 35 |
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image::https://i.nostr.build/kuB6esNcMRElqAWt.png[Figure 4, 300] |
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|
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until the mid to late 30 s. Since 2010, first birth rates have decreased among both cohabiting and married couples for nearly all ages, but more distinctly at younger ages. The decrease was, however, less pronounced among married women. For instance, from 2010 through 2018, the first birth rate at age 25 decreased from 0.27 to 0.21 among married women and from 0.09 to 0.06 among cohabiting women. Furthermore, the low probability of transitioning from single to first birth—specifically the transition from single to first birth and entering a union—has decreased. We observed a sharp increase in the probability of remaining either as a cohabiting or a married couple without transitioning to a first birth. |
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|
||||
4.2 Transition Probabilities by Socioeconomic Status (SES) Groups |
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|
||||
We further explored whether changes in partnering and first birth transitions were more pronounced in some SES groups than others (Fig. 4). We show the results for the lowest and highest SES groups. |
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|
||||
The decrease in the transition to cohabitation observed at younger ages in the general population was visible primarily among less educated grosups of women and men. Over years 2010—2018, the probability of forming a cohabiting union at age 25 declined for primary educated men and women by 5 and 3 percentage points, respectively. This is a 45% (men) and a 27% Fig. 5 Contributions of declining first births, changes in union formations, and changes in union dissolutions to the decline in the percentage experiencing first births based on the first birth rates in 2010–2018. The black solid line indicates the percentage experiencing a first birth that would have been observed if the population in 2010 would have experienced the 2010 transition rates in the years 2010 through 2018. Shaded areas indicate how much the decline in first births would have been dampened if the corresponding transition probabilities would not have changed (women) decline of the levels observed in year 2010. We observed no change in cohabitation rates among women |
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|
||||
.Contributions of declining first births, changes in union formations, and changes in union dissolutions to the decline in the percentage experiencing first births based on the first birth rates in 2010–2018. The black solid line indicates the percentage experiencing a first birth that would have been observed if the population in 2010 would have experienced the 2010 transition rates in the years 2010 through 2018. Shaded areas indicate how much the decline in first births would have been dampened if the corresponding transition probabilities would not have changed |
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image::https://i.nostr.build/q2xoB3Jnk6UPkHnH.png[Figure 5, 300] |
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|
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with tertiary-level education, but found a decline (6 percentage point which is 30% of the level in 2010) for the relatively small group of higher tertiary-level educated 25-year-old men. Furthermore, a long-term decreasing trend in the transition from cohabitation to marriage was observed across all SES groups, but appeared somewhat stronger among the more highly educated. For instance, the transition probability from cohabitation to marriage at age 25 remained low at around 0.05 for primary educated women, but fell from 0.18 to 0.10 for higher tertiary educated women. |
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|
||||
We also observed a slight increase in the transition from cohabitation to being single primarily among the least educated men and women. Moreover, we observed no change in the transition from marriage to being single in any of the educational groups, except for a small potential increase among the least educated at younger ages. |
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|
||||
Finally, we find that first birth rates have decreased both among cohabiting and married men and women across all SES groups. However, the decrease was less pronounced in the small group of married women in the lowest educational group. |
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|
||||
4.3 Contributions to Declining First Births, 2010–2018 |
||||
|
||||
Figure 5 shows the contributions of the changes in first birth transitions, union formations, and union dissolutions to the decline in the number of first births from 2010 through 2018. We use the percentage experiencing a first birth, a synthetic age-standardised measurement indicating the proportion that expected to experience a first birth based on the observed rates. The observed changes in the transition probabilities led to a decline in the share experiencing a first birth to 68.1% for women and to 58.4% for men. This natural course scenario matches well with the true observed first birth rates (Appendix Fig. 10). If the transition probabilities remained stable (i.e. as observed in 2010) across years, the share experiencing a first birth based on the age-specific first birth rates in 2018 would have reached 78.6% for women and 71.2% for men. |
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|
||||
If first birth transitions (whether among single, cohabitating, or married individuals) had not decreased, 76% of the observed decline in the share experiencing a first birth among women would have been dampened. The largest contributions originated from cohabiting women (42%), followed by women who were single at the end of the year, but who experienced their first birth and lived in a union at the end of the following year (17%) and married women (13%). The decrease in single motherhood also insubstantially contributed (4%). Furthermore, if union formation had remained stable, the decrease would have been dampened by 21%. Decreasing marriage rates appeared vastly more important (19%) than decreasing cohabitating rates (2%). In addition, the small increase in dissolution rates contributed modestly (6%). It is also possible that the couple did not marry because they did not have the first birth in the first place. Among men, the results were largely similar, but, the decrease in cohabitation rates, for instance, was slightly more important among men than among women. |
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|
||||
4.3.1 Contribution by Socioeconomic Status Groups |
||||
|
||||
The results of the decomposition are affected by both the probabilities of transitioning between different states, as well as the population composition, which both differ between SES groups and contribute partially to findings between groups. In order to more intuitively grasp the compositional effects of transitions between states in the counterfactual simulation, we show the distribution of union states in 2009 and 2017 for the lowest and highest educational groups in (Appendix Fig. 8). Notably, the proportion single is much higher among the lower educated, and more people are cohabitating and married among the higher educated, throughout the study period. When it comes to changes in the last decade, we observe that among the lower educated, the proportion single has increased while the proportion in unions (both cohabiting and married) has decreased. Among the higher educated, the proportion single has remained relatively stable, but the proportion married decreased and the proportion cohabiting increased. |
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|
||||
Figure 6 shows the contributions to the first birth decline based on education. The total decline was larger in the lower SES groups. Specifically, the share ever experiencing a first birth fell from 2010 through 2018 from 65 to 48% for women with primary education and from 82 to 75% for women with higher tertiary education. In addition, we noted that the contribution associated with changes in first birth transitions versus changes in unions to the declining first birth rate is, in general, similar across all SES groups. However, some differences emerged. First, the decline in first birth transitions among married individuals explained a larger share of the total first birth decline among women with a higher level of education. That is, the share |
||||
|
||||
.Contributions of declining first births, changes in union formations, and changes in union dissolutions to the decline in th percentage experiencing first births based on the first birth rates in 2010–2018 by education groups. The top curve shows the results for the higher tertiary education groups, while the bottom curve shows the results for the primary-level education groups |
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image::https://i.nostr.build/F6K02I9QC7ONbUh9.png[Figure 6, 300] |
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|
||||
explained is almost four times higher among those with a tertiary-level education (27%) than among those who completed primary education alone (7%). This reflects both the fact that the first birth transition probabilities for married individuals with primary-level education declined less as well as the difference in the population composition: a much higher proportion of more highly educated individuals were married (Appendix Fig. 8). In turn, the declining transition to single motherhood appeared more important for women with the least education. |
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|
||||
Turning to union formation, declining cohabitation rates appear more important among women with the least education. The contribution of changes in cohabitation rates negatively (although slightly) associated with a tertiary level of education among women, meaning that cohabitation rates have, in fact, increased among this group. If such rates remained unchanged, first birth rates would have decreased slightly faster. This also results from the low number of more highly educated individuals at ages coinciding with decreases in the cohabitation rates. Here, again, the results for men are, in general, rather similar, yet some differences exist. For instance, the declining cohabitation rates among the least educated were slightly more important among men. In addition, the contribution of decreasing single parenthood did not concentrate within any educational group as they did among women. |
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|
||||
4.4 Summary of the Results |
||||
|
||||
We observed a lower fertility in unions after 2010, a long-term decline in marriage rates and increasing dissolution rates among cohabiting couples. Lower fertility in unions explained around three-quarters of the total decline in first births, and the decreasing first birth transitions appeared more important among cohabiting couples than among married individuals. Consequently, changes in unions explained around one-quarter of the total decline in first births. Furthermore, lower marriage rates were more important than changes in cohabitation formation and cohabitation dissolution, and increases in cohabitation dissolution were more important than declining figures in cohabitation formation in explaining the first birth decline. Results were similar for both men and women. |
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|
||||
Our findings were largely consistent across SES groups, whereby first births declined and childbearing within unions explained most of this decline in all groups. In agreement with the study by Comolli et al. (2020), the total decline in first births was stronger among less educated groups. Nevertheless, the decrease in the transition from cohabitation to marriage explained more of the decline among the most highly educated, while the decrease in the transition from single to cohabitation explained more of the decline among the least educated. We also compared the findings among income quartile groups as a robustness check (Appendix Figs. 9, 11): results remained relatively consistent regardless of whether education or income was used as an indicator of socioeconomic position. We note, however, that some differences between income groups were not as strong as those that emerged between educational groups. Moreover, while the probability of experiencing a first birth among cohabiting couples declined rather consistently across SES groups regardless of the measure, among married couples the decline was weaker among the least educated, potentially reflecting the selection of this small group. |
||||
|
||||
== 5 Discussion |
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|
||||
This study investigated how the fertility decline in the 2010s in Finland associated with changes in fertility in unions and with changes in union formation and dissolution. Fertility mainly occurs in unions (Jalovaara & Fasang, 2017; Kiernan, 1999), and pinning down whether the fertility decline occurs within unions or because lack of unions is to zoom on the potential causes of the fertility decline. Using full-coverage Finnish register data, an incidence-based multistate model, and a counterfactual approach, we analysed the changes in the transition probabilities between relationship states among single, cohabitating, and married individuals, and in the transition to first birth among each relationship state in Finland from 2000 through 2018, and estimated the impact of these changes on the first birth decline in Finland during the 2010s. While patterns in unions explained some of the decline, the decreased transition to the first birth within unions mattered more. |
||||
|
||||
First births in contemporary Finland have increasingly been born to cohabiting couples, whereby marriage has increasingly followed childbearing. In this current trend, however, the declining marriage rates since 2010 are not followed by increasing nonmarital births, as the decreases in first births among cohabiting couples represent the primary driver of declines in total first births. Instead, the tendency to remain cohabiting without transitioning to either marriage or the first birth has increased rapidly, accompanied by a slight increase in the risk of separation among younger cohabiting couples. A slower decline in the first birth transition among married as compared to cohabiting couples was expected, since marriage reflects a stronger commitment and promotes childbearing more strongly than cohabitation (Jalovaara & Miettinen, 2013; Miettinen & Rotkirch, 2008). It may be that childbearing intentions have declined among cohabiting and married couples in Finland. Finnish surveys indicate that the proportion voluntary childless has risen in Finland: the probability of having childfree ideals at age 25 was around 5% for the 1975–79 female cohort and around 20% for the 1990–1994 female cohort (Savelieva et al., 2021a, 2021b). |
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|
||||
It also seems plausible that weaker childbearing intentions may have contributed to declining rates of union formation: hence, it might be that the lack of a first birth itself contributed to the decline in marriage. Marriage is strongly related to the family formation process so that couples who (wish to) have children often marry, and vice versa, marriage promotes the first birth (Brien et al., 1999; Steele et al., 2005). Consequently, the first birth decline attributed to declining marriage rates does not necessarily reflect a tendency to eventually marry less (unless the couples remain childless). Given that in the Nordic countries marriage increasingly occurs after the first birth rather than before (Jónsson, 2020), to better understand the causality of declining marriage and declining first births, future studies could investigate marriage patterns among parents as well. |
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|
||||
The changes in family formation witnessed since the 1960s in the Nordic and other countries, such as postponing and declining fertility and the rise in alternative living arrangements compared to marriage, have often been explained by the second demographic transition theory (SDT), where the weakening of traditional family values gives rise to individual autonomy and self-actualisation (Lesthaeghe, 2014). Contrary to the predictions of SDT, the recent new family demographic developments in Finland were not necessarily driven by more highly educated individuals. Perhaps in Finland—a country with a long history of individualistic values (Sobotka, 2008)—these values have already spread from the higher social strata to the whole population by the 2010s. Interestingly, a characteristic SDT feature, voluntary childlessness, seems to have gained ground in Finland only in the last decade (Savelieva et al., 2021b) and may explain some of the fertility decline. |
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|
||||
Increasing uncertainty that goes beyond the actual own economic circumstances has been suggested to drive the recent declining fertility trends in the Nordic (Comolli et al., 2020) and other European countries (Vignoli et al., 2020a, 2020b). The patterns we observed in Finland agree to some extent with this view. First, the declining fertility rates in unions explained the lion’s share of the decline in first births. This could be viewed as in line with the reasoning on uncertainty, given that uncertainty is less likely to be an obstacle to forming a cohabiting union than it is for more permanent and irreversible life decision like childbearing or marriage (Guetto et al., 2020). Also Finnish surveys suggest that perceived uncertainty is considered the most important subjective reason to postpone childbearing in the 2010s (Savelieva et al., 2021a). Second, the patterns remained largely consistent across SES groups, which would be generally not discrepant with the view of uncertainty. The sharper decline among the lower SES groups, as found in this study, however, suggests that actual economic constraints may not be completely irrelevant for the recent changes. |
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|
||||
Although the decrease in the transition to cohabitation did not strongly explain the total decline in first births, its role should not be dismissed. First births remained rather unaffected by decreased cohabitation rates at younger ages, much due to the fact that first birth rates among young cohabitating couples are low. Moreover, a declining trend towards entering a cohabiting union remained notable primarily among the lower SES groups. Cohabitation rates have been stable in Finland for quite some time, but began declining at younger ages after 2015, a departure from the previous long-term trend. It is still unclear whether this merely reflects postponement or if we are also observing an increase in the share of individuals who never partner in the near future. The sharper decline in cohabitation rates among the lower SES groups, particularly among men, may imply that these groups are experiencing greater difficulties in the mating market. We agree with previous views arguing that the increase in the availability of dating partners through online dating sites and its effect on (un)stable union formation is an avenue for further investigation (Hiilamo, 2020). |
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|
||||
If future transition probabilities remain at current levels and no catching up occurs, an inevitable consequence of the current first birth decline will result in increasing levels of ultimate childlessness. The expected proportion of childless individuals in Finland based on the first birth rates in 2010 and 2018, respectively, increased from 21.0% to 31.5% among women and from 28.7% to 41.5% among men. Given the sharp decline in childbearing within unions, future ultimate childlessness could potentially be related less to the absence of unions. Previously, ultimate childlessness in Finland has been linked to never partnering and to short or unstable spells of cohabitation (Jalovaara & Fasang, 2017), and rates of ultimate childlessness are low among married couples (Saarela & Skirbekk, 2019). Our findings suggest that all educational groups may witness increases in the rates of ultimate childlessness due to a declining progression to first births within unions alongside greater union instability, and the lower SES groups additionally due to the increasing difficulties in forming unions. |
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|
||||
The primary strength of this study lies in our use of full-population Finnish register data, which distinguished nonmarital cohabitation from marriage. This distinction is crucial for understanding family-specific demographic changes within a context where cohabitation typically represents the first step in the family formation process and where first births are increasingly concentrated within cohabitating unions. A further strength lies in our analyses of data for both men and women, since analyses on men’s fertility remain rare. We also acknowledge several limitations in studying family formation based on educational attainment in the current period perspective, where the share enrolled (and in the decomposition analysis also the age structure) varies between groups with varying levels of attainment. Our sensitivity analysis (Appendix Fig. 12) revealed that the primary results regarding changes over time were quite similar regardless of whether we included currently enrolled or not. Reassuringly, the analysis by income group also indicated largely similar socioeconomic patterns in recent declines as those found from our analysis based on education. |
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|
||||
Our model utilises annual transitions, given that it represents a straightforward way of proceeding and because personal information is available at the end of each calendar year. Potentially, using shorter transition periods for cohabitation, such as every three months, might prove appropriate, since that is the minimum time within which to capture unions in the registers. But, such short transitions may not be more informative vis-à-vis first birth transitions. An important question arises regarding whether we missed short-term changes in our approach. For instance, it is possible to observe a decline in the annual transition to cohabitation simply if the formation of cohabiting unions lasting shorter than one year increases while the total number of new cohabitations remains constant. Additional analyses revealed that the formation of unions lasting shorter than one year has also decreased in recent years (Appendix Table 2). In addition, our model did not take into account the duration of unions (Hoem et al., 2013; Jalovaara & Kulu, 2018). That said, additional analyses revealed that first birth rates have decreased, and separation rates have increased rather similarly regardless of union length (Appendix Fig. 13). |
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|
||||
In conclusion, this study demonstrated that the sharp decline in first birth rates in the 2010s in Finland is associated with changes in partnering. However, the declining tendency to experience a first birth within unions is most important in explaining the fertility decline. Future studies should specifically focus on the declining tendency to transition to parenthood among cohabiting couples, as well as to the increasing instability of such unions. |
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|
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== Appendix |
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|
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See Figs. 7, 8, 9, 10, 11, 12, 13 and Tables 1 and 2. |
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|
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image::https://i.nostr.build/hk9f6K8nJsIbe51b.png[Figure 7-1, 300] |
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.Age-specific transition probabilities for single, cohabitating, and married individuals, and the first birth among childless women and men in Finland, 2001–2018 |
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image::https://i.nostr.build/uzUGdpxaviBtjQ80.png[Figure 7-2, 300] |
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|
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.First birth rates (births per childless woman) based on the Human Fertility Database (HFD) and the natural course in 2010 and 2018 |
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image::https://i.nostr.build/j9GSlPz6e7RbPaQX.png[Figure 8, 300] |
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|
||||
.The population by partnership state in 2009 and 2017, for men and women with primary and higher tertiary education |
||||
image::https://i.nostr.build/AB6lepAt4cp9rGCU.png[Figure 9, 300] |
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|
||||
.Transition probabilities for single, cohabitating, and married individuals, and first births among childless women and men in 2001–2018 by income quartile group at ages 25 and 35 |
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image::https://i.nostr.build/SOFrUTSM1oyJ4vJ7.png[Figure 10, 300] |
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|
||||
image::https://i.nostr.build/H1JoftCfVic50vgp.png[Figure 11-1, 300] |
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image::https://i.nostr.build/iN9gSeNpyBooqTpN.png[Figure 11-2, 300] |
||||
.Contributions of declining first births, changes in union formations, and changes in union dissolutions to the decline in the percentage experiencing first births based on the first birth rates in 2010– 2018 by income quartile groups. The top curve shows the results for the highest income group, while the bottom curve shows the results for the lowest income group |
||||
image::https://i.nostr.build/2XENJrAEb8MAB3zD.png[Figure 11-3, 150] |
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|
||||
image::https://i.nostr.build/mHe0PJHGqUkMhQmc.png[Figure 12-1, 300] |
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image::https://i.nostr.build/LScumIjd0TI4QiBS.png[Figure 12-2, 300] |
||||
.Contributions of declining first births, changes in union formations, and changes in union dissolutions to the decline in the percentage experiencing first births based on the first birth rates in 2010– 2017 by education groups, excluding students. (Information regarding education enrolment was incomplete in 2018). The top curve shows the results for the most educated, whilst the bottom curve shows the results for the least educated |
||||
image::https://i.nostr.build/2XENJrAEb8MAB3zD.png[Figure 12-3, 150] |
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|
||||
.Transition probabilities by length of union for women aged < 25, 25–29 and 30–34, in 2010 and 2018. Length of union measured at the end of the previous year |
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image::https://i.nostr.build/TEJwnLyLrOuTkT1r.png[Figure 13, 300] |
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|
||||
.Table 1 Number of first births, person-years of exposures, and partnership status transitions among the childless population, ages 15 to 44 among women and men, 2000–2018 |
||||
[cols="2,>1,>1,>1", stripes] |
||||
|=== |
||||
|Description|All|Women|Men |
||||
|
||||
|Characteristic|(n = 2 125 172)|(n = 993 024)|(n = 1 132 148) |
||||
|
||||
|First births, number|740 537|379 051|361 486 |
||||
|
||||
|Person-years of follow-up|19 468 815|8 806 785|10 662 030 |
||||
|
||||
4+e|Partnership transitions, number |
||||
|
||||
|Single to single|12 419 504|5 232 453|7 187 051 |
||||
|
||||
|Single to cohabitating|1 206 555|604 671|601 884 |
||||
|
||||
|Single to married|102 370|51 424|50 946 |
||||
|
||||
|Single to first birth & single|47 249|31 398|15 851 |
||||
|
||||
|Single to first birth & union|86 883|40 965|45 918 |
||||
|
||||
|Cohabitating to single|536 569|268 414|268 155 |
||||
|
||||
|Cohabitating to cohabitating|3 018 824|1 532 222|1 486 602 |
||||
|
||||
|Cohabitating to married|269 197|136 150|133 047 |
||||
|
||||
|Cohabitating to first birth|334 546|168 932|165 614 |
||||
|
||||
|Married to single|47 898|24 330|23 568 |
||||
|
||||
|Married to cohabitating|8 417|4 670|3 747 |
||||
|
||||
|Married to married|1 118 944|573 400|545 544 |
||||
|
||||
|Married to first birth|271 859|137 756|134 103 |
||||
|=== |
||||
|
||||
.Table 2 Number of unions by starting year and union length, women aged < 25, 2000–2018 |
||||
[cols="1,>1,>1,>1,>1,>1", stripes] |
||||
|=== |
||||
|Year|>=1 year|[274–365) days|[183–274) days|[90–183) days|Total |
||||
|
||||
|2000|20,014|1191|1051|927|23,183 |
||||
|
||||
|2001|20,755|1213|1095|838|23,901 |
||||
|
||||
|2002|19,937|1215|1040|835|23,027 |
||||
|
||||
|2003|19,827|1254|1035|857|22,973 |
||||
|
||||
|2004|20,672|1143|1098|821|23,734 |
||||
|
||||
|2005|20,543|1219|1071|838|23,671 |
||||
|
||||
|2006|20,298|1247|1082|782|23,409 |
||||
|
||||
|2007|20,001|1239|1041|765|23,046 |
||||
|
||||
|2008|20,334|1273|1047|778|23,432 |
||||
|
||||
|2009|20,526|1215|1108|745|23,594 |
||||
|
||||
|2010|19,942|1267|1073|789|23,071 |
||||
|
||||
|2011|20,243|1363|1092|801|23,499 |
||||
|
||||
|2012|20,392|1296|1119|866|23,673 |
||||
|
||||
|2013|20,115|1341|1095|804|23,355 |
||||
|
||||
|2014|19,211|1260|1104|832|22,407 |
||||
|
||||
|2015|19,042|1237|1069|829|22,177 |
||||
|
||||
|2016|18,038|1326|1128|734|21,226 |
||||
|
||||
|2017|15,886|1095|859|652|18,492 |
||||
|
||||
|2018|15,969|1074|889|643|18,575 |
||||
|=== |
||||
|
||||
== Supplementary Information |
||||
|
||||
The online version contains supplementary material available at https://doi.org/10.1007/s10680-022-09605-8. |
||||
|
||||
Acknowledgements |
||||
|
||||
This research was funded by the Strategic Research Council (SRC), FLUX consortium, decision numbers: 345130 and 345131. The Academy of Finland supported the contributions of Julia Hellstrand (Project No. 294861) and Jessica Nisén (Project No. 332863 and 320162). The authors would like to acknowledge Tim Riffe and Christian Dudel for help with the methods. |
||||
|
||||
Authors’ contributions |
||||
|
||||
All authors contributed to the study conception and design. Data analysis was performed by Julia Hellstrand. The first draft of the manuscript was written by Julia Hellstrand, and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript. |
||||
|
||||
Funding |
||||
|
||||
Open Access funding provided by University of Helsinki including Helsinki University Central Hospital. |
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|
||||
Data Availability |
||||
|
||||
Individual level register data from Statistics Finland are not freely available. |
||||
|
||||
Code Availability |
||||
|
||||
The code is available upon request. |
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|
||||
Declarations |
||||
|
||||
Conflicts of interest |
||||
|
||||
The authors declare that there is no conflict of interest. |
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|
||||
Open Access |
||||
|
||||
This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. |
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|
||||
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Mills, M., & Blossfeld, H.-P. (2003). Globalization, uncertainty and changes in early life courses. Zeitschrift für Erziehungswissenschaft, 6, 188–218. |
||||
|
||||
Mills, M., and Blossfeld, H.-P. (2013). The second demographic transition meets globalization: A comprehensive theory to understand changes in family formation in an era of rising uncertainty. In Ann Evans and Janeen Baxter (eds.), Negotiating the life course: Stability and change in life pathways. Springer. |
||||
|
||||
Mills, M., Rindfuss, R. R., McDonald, P., & te Velde, E. (2011). Why do people postpone parenthood? Reasons and social policy incentives. Human Reproduction Update, 17, 848–860. |
||||
|
||||
Myrskylä, M., Billari, F. C. and Kohler, H. P. (2011). High development and fertility: Fertility at older reproductive ages and gender equality explain the positive link, Max Planck Institute for Demographic Research, Rostock, Germany. (MPIDR Working Paper WP-2011–017). |
||||
|
||||
Myrskylä, M., Kohler, H. P., & Billari, F. C. (2009). Advances in development reverse fertility declines. Nature, 460, 741–743. |
||||
|
||||
Mäenpää, E., & Jalovaara, M. (2015). Achievement replacing ascription? Changes in homogamy in education and social class origins in Finland. Advances in Life Course Research, 26, 76–88. |
||||
|
||||
Nisén, J., Martikainen, P., Myrskylä, M., & Silventoinen, K. (2018). Education, other socioeconomic characteristics across the life course, and fertility among finnish men. European Journal of Population, 34, 337–366. |
||||
|
||||
Nisén, J., Martikainen, P., Silventoinen, K., & Myrskylä, M. (2014). Age-specific fertility by educational level in the Finnish male cohort born 1940–1950. Demographic Research, 31, 119–136. |
||||
|
||||
OECD. (2021a). Education at a glance 2021. |
||||
|
||||
OECD. (2021b). OECD family database [electronic resource], Accessed 10.8.2021. http:// www. oecd. org/ social/ family/ datab ase. htm. |
||||
|
||||
Official Statistics of Finland (OSF). (2018a). Births [e-publication]., Helsinki: Statistics Finland, Accessed 21.1.2019. https://www.stat.fi/til/synt/2017/synt_2017_2018-04-27_tie_001_en.html. |
||||
|
||||
Official Statistics of Finland (OSF). (2018b).Changes in marital status [e-publication], Helsinki: Statistics Finland, Accessed 7.5.2020. https://www.stat.fi/til/ssaaty/2018/saaty_2018_2019-06-18_kuv_001_en.html. |
||||
|
||||
Official Statistics of Finland (OSF). (2019). Decrease in birth rate slowed down in 2019, Helsinki: Statistics Finland, Accessed 31.8.2020. 2171 3 |
||||
Less Partnering, Less Children, or Both? Analysis of the Drivers… |
||||
Supplementary Information |
||||
|
||||
Official Statistics of Finland (OSF). (2021). Families [e-publication], Helsinki: Statistics Finland, Accessed 11.3.2021. http://www.stat.fi/til/perh/kas_en.html. |
||||
|
||||
Perelli-Harris, B., Sigle-Rushton, W., Kreyenfeld, M., Lappegård, T., Keizer, R., & Berghammer, C. (2010). The educational gradient of childbearing within cohabitation in Europe. Population and Development Review, 36, 775–801. |
||||
|
||||
Rotkirch, A. (2020). The wish for a child, Vienna Yearbook of Population Research 2020, 18. first online: 25.11.2020. |
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|
||||
Rotkirch, A., and Miettinen, A. (2017). Childlessness in Finland. In Michaela Kreyenfeld and Dirk Konietzka (Eds.), Childlessness in Europe: Contexts, causes, and consequences. Springer. |
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||||
Saarela, J., and Skirbekk, V. (2019). Childlessness and union histories: Evidence from Finnish population register data. Journal of Biosocial Science: 1–19. |
||||
|
||||
Savelieva, K., Jokela, M., & Rotkirch, A. (2021a). Reasons to postpone or renounce childbearing during fertility decline in Finland. SocArXiv. |
||||
|
||||
Savelieva, K., Nitsche, N., Berg, V., Miettinen, A., Rotkirch, A., & Jokela, M. (2021b). Birth cohort changes in fertility ideals: Evidence from repeated cross-sectional surveys in Finland. SocArXiv. Sobotka, T. (2008). Overview chapter 6: The diverse faces of the second demographic transition in Europe. Demographic Research, S7, 171–224. |
||||
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||||
Sobotka, T. (2017). Childlessness in Europe: Reconstructing long-term trends among women born in 1900–1972. In Michaela Kreyenfeld and Dirk Konietzka (Eds.), Childlessness in Europe: Contexts, causes, and consequences. Springer. |
||||
|
||||
Sobotka, T., Skirbekk, V., & Philipov, D. (2011). Economic recession and fertility in the developed world. Population and Development Review, 37, 267–306. |
||||
|
||||
Steele, F., Kallis, C., Goldstein, H., & Joshi, H. (2005). The relationship between childbearing and transitions from marriage and cohabitation in Britain. Demography, 42, 647–673. |
||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
Zeman, K., Beaujouan, É., Brzozowska, Z., & Sobotka, T. (2018). Cohort fertility decline in low fertility countries: Decomposition using parity progression ratios. Demographic Research, 38, 651–690. |
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|
||||
= Lorem Ipsum |
||||
|
||||
== Asciidoc test |
||||
|
||||
Here are some nostr addresses: |
||||
|
||||
nostr:naddr1qppkw6t5vd5hgctyv4kz6urjda4x2cm594jx7cm4d4jkuarpw35k7m3dva5hgcmfw3skgetv94c8ymm2v43hgtf39438jttnw3jkcmrp94mz6vgprdmhxue69uhhg6r9vd5hgctyv4kzumn0wd68yvfwvdhk6q3q2umrfdjgvdxt45g0y3ghwcyfagssjrv5qlm3t6pu2aa5vydwdmwqxpqqqp64jqj69ta |
||||
|
||||
nostr:npub149p5act9a5qm9p47elp8w8h3wpwn2d7s2xecw2ygnrxqp4wgsklq9g722q |
||||
|
||||
nostr:nprofile1qyghwumn8ghj7mn0wd68ytnhd9hx2tcpzamhxue69uhhyetvv9ujumn0wd68ytnzv9hxgtcqyz55xnhpvhksrv5xhm8uyac779c96dfh6pgm8peg3zvvcqx4ezzmu4qdkdl |
||||
|
||||
nostr:naddr1qvzqqqr4gupzp22rfmsktmgpk2rtan7zwu00zuzax5maq5dnsu5g3xxvqr2u3pd7qyghwumn8ghj7mn0wd68ytnhd9hx2tcpzamhxue69uhhyetvv9ujumn0wd68ytnzv9hxgtcqp4axzurrdpshger9wd5kwmscmk4nl |
||||
|
||||
nostr:nevent1qvzqqqqqqypzp22rfmsktmgpk2rtan7zwu00zuzax5maq5dnsu5g3xxvqr2u3pd7qyghwumn8ghj7mn0wd68ytnhd9hx2tcpzamhxue69uhhyetvv9ujumn0wd68ytnzv9hxgtcqyrz6y4vq9the6z78t0307nrvf3kz6m6jcu9ma6defzg6yc02tn2vqsdp3f7 |
||||
|
||||
nostr:note1ck39tqp2a7wsh36mutl5cmzvdskk75k8pwlwnw2gjx3xr6ju6nqqcald30 |
||||
|
||||
|
||||
image::https://i.pinimg.com/474x/bd/02/21/bd022104b73d5e5c1f0cccfd9892dff2.jpg[Woman reading] |
||||
|
||||
[.lead] |
||||
Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo _duo dolores et ea rebum_. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. |
||||
|
||||
.Important quote |
||||
.... |
||||
Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt *ut labore et dolore* magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. |
||||
.... |
||||
|
||||
.List of ordered items |
||||
. first item |
||||
. second item |
||||
.. second subitem 1 |
||||
.. second subitem 2 |
||||
. third item |
||||
|
||||
.List of unordered items |
||||
* item |
||||
* item |
||||
** a subitem |
||||
*** a sub-subitem |
||||
|
||||
Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. |
||||
|
||||
.Resized image with wrapped text |
||||
image::https://i.pinimg.com/736x/2b/d8/05/2bd805691ef4d47b28dc0a7bf5ab7dbc.jpg[tennis players, 300, , float="right", align="center"] |
||||
|
||||
Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. |
||||
|
||||
[source, python] |
||||
---- |
||||
# This program adds two numbers |
||||
|
||||
num1 = 1.5 |
||||
num2 = 6.3 |
||||
|
||||
# Add two numbers |
||||
sum = num1 + num2 |
||||
|
||||
# Display the sum |
||||
print('The sum of {0} and {1} is {2}'.format(num1, num2, sum)) |
||||
---- |
||||
|
||||
Duis autem vel eum iriure dolor in hendrerit in vulputate velit esse molestie consequat, vel illum dolore eu feugiat nulla facilisis at vero eros et accumsan et iusto odio dignissim qui blandit praesent luptatum zzril delenit augue duis dolore te feugait nulla facilisi. https://www.loremipsum.de/[Lorem ipsum] dolor sit amet, consectetuer adipiscing elit, sed diam nonummy nibh euismod tincidunt ut laoreet dolore magna aliquam erat volutpat. |
||||
|
||||
INFO: This should be formatted as a single-paragraph admonition. Ut wisi enim ad minim veniam, quis nostrud exerci tation ullamcorper suscipit lobortis nisl ut aliquip ex ea commodo consequat. Duis autem vel eum iriure dolor in hendrerit in vulputate velit esse molestie consequat, vel illum dolore eu feugiat nulla facilisis at vero eros et accumsan et iusto odio dignissim qui blandit praesent luptatum zzril delenit augue duis dolore te feugait nulla facilisi. |
||||
|
||||
Nam liber tempor cum soluta nobis eleifend option congue nihil imperdiet doming id quod mazim placerat facer possim assum. Lorem ipsum dolor sit amet, consectetuer adipiscing elit, sed diam nonummy nibh euismod tincidunt ut laoreet dolore magna aliquam erat volutpat. Ut wisi enim ad minim veniam, quis nostrud exerci tation ullamcorper suscipit lobortis nisl ut aliquip ex ea commodo consequat. |
||||
|
||||
[WARNING] |
||||
.This is a block admonition |
||||
==== |
||||
Duis autem vel eum iriure dolor in hendrerit in vulputate velit esse molestie consequat, vel illum dolore eu feugiat nulla facilisis. |
||||
|
||||
At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, At accusam aliquyam diam diam dolore dolores duo eirmod eos erat, et nonumy sed tempor et et invidunt justo labore Stet clita ea et gubergren, kasd magna no rebum. sanctus sea sed takimata ut vero voluptua. est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat. |
||||
==== |
||||
|
||||
Consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. |
||||
|
||||
.Try out an MP3 link |
||||
audio::https://www.gutenberg.org/files/3002/3002-h/mp3/sochi-med.mp3[audio file] |
||||
|
||||
.And a video link |
||||
video::https://www.youtube.com/watch?v=9aqVxNCpx9s[Moldau] |
||||
|
||||
.And a youtube link |
||||
video::9aqVxNCpx9s[youtube] |
||||
|
||||
Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus. |
||||
|
||||
.Windtrainer workouts |
||||
[width="80%",cols="3,^2,^2,10",options="header"] |
||||
|==== |
||||
|Date |Duration |Avg HR |Notes |
||||
|
||||
|22-Aug-08 |10:24 | 157 | |
||||
Worked out MSHR (max sustainable heart rate) by going hard |
||||
for this interval. |
||||
|
||||
|22-Aug-08 |23:03 | 152 | |
||||
Back-to-back with previous interval. |
||||
|
||||
|24-Aug-08 |40:00 | 145 | |
||||
Moderately hard interspersed with 3x 3min intervals (2min |
||||
hard + 1min really hard taking the HR up to 160). |
||||
|
||||
|==== |
||||
|
||||
And that is all. |
||||
|
||||
== Plain-text below here |
||||
|
||||
Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. |
||||
|
||||
Duis autem vel eum iriure dolor in hendrerit in vulputate velit esse molestie consequat, vel illum dolore eu feugiat nulla facilisis at vero eros et accumsan et iusto odio dignissim qui blandit praesent luptatum zzril delenit augue duis dolore te feugait nulla facilisi. Lorem ipsum dolor sit amet, consectetuer adipiscing elit, sed diam nonummy nibh euismod tincidunt ut laoreet dolore magna aliquam erat volutpat. |
||||
|
||||
Ut wisi enim ad minim veniam, quis nostrud exerci tation ullamcorper suscipit lobortis nisl ut aliquip ex ea commodo consequat. Duis autem vel eum iriure dolor in hendrerit in vulputate velit esse molestie consequat, vel illum dolore eu feugiat nulla facilisis at vero eros et accumsan et iusto odio dignissim qui blandit praesent luptatum zzril delenit augue duis dolore te feugait nulla facilisi. |
||||
|
||||
Nam liber tempor cum soluta nobis eleifend option congue nihil imperdiet doming id quod mazim placerat facer possim assum. Lorem ipsum dolor sit amet, consectetuer adipiscing elit, sed diam nonummy nibh euismod tincidunt ut laoreet dolore magna aliquam erat volutpat. Ut wisi enim ad minim veniam, quis nostrud exerci tation ullamcorper suscipit lobortis nisl ut aliquip ex ea commodo consequat. |
||||
|
||||
Duis autem vel eum iriure dolor in hendrerit in vulputate velit esse molestie consequat, vel illum dolore eu feugiat nulla facilisis. |
||||
|
||||
At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, At accusam aliquyam diam diam dolore dolores duo eirmod eos erat, et nonumy sed tempor et et invidunt justo labore Stet clita ea et gubergren, kasd magna no rebum. sanctus sea sed takimata ut vero voluptua. est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat. |
||||
|
||||
Consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus. |
||||
|
||||
Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. |
||||
|
||||
Duis autem vel eum iriure dolor in hendrerit in vulputate velit esse molestie consequat, vel illum dolore eu feugiat nulla facilisis at vero eros et accumsan et iusto odio dignissim qui blandit praesent luptatum zzril delenit augue duis dolore te feugait nulla facilisi. Lorem ipsum dolor sit amet, consectetuer adipiscing elit, sed diam nonummy nibh euismod tincidunt ut laoreet dolore magna aliquam erat volutpat. |
||||
|
||||
Ut wisi enim ad minim veniam, quis nostrud exerci tation ullamcorper suscipit lobortis nisl ut aliquip ex ea commodo consequat. Duis autem vel eum iriure dolor in hendrerit in vulputate velit esse molestie consequat, vel illum dolore eu feugiat nulla facilisis at vero eros et accumsan et iusto odio dignissim qui blandit praesent luptatum zzril delenit augue duis dolore te feugait nulla facilisi. |
||||
@ -0,0 +1,351 @@
@@ -0,0 +1,351 @@
|
||||
= Sharing with you the mystical truth of the Rauhnächte |
||||
|
||||
== The First Rauhnacht |
||||
|
||||
December 24th to 25th |
||||
|
||||
image::https://i.nostr.build/05pLUfFRKTcMQBGr.jpg[1st Rauchnacht, 300] |
||||
|
||||
.The first mystical night of twelve |
||||
[%hardbreaks] |
||||
Month: *January* |
||||
Namesake: *Anastasia* |
||||
Theme: *From darkness into light* |
||||
|
||||
The first Rauhnacht is dedicated to the month of January, carrying the energy of Anastasia, "the one who rises." It symbolizes emerging from darkness into light and invites reflection on your foundation. Strengthen your roots—what is the state of your earthly grounding, and what does it rest upon? |
||||
|
||||
* Honor and bless your roots while contemplating what aspects of your life or lineage need healing. |
||||
* Consider what is needed to find peace and balance. |
||||
* Remember, true prosperity (Wohlstand) means "to stand well" (wohl stehen). |
||||
* Light a candle for your ancestors and carry its glow into your dreams. |
||||
* Leave a gift for nature at the roots of a tree, offering it along with good wishes to your ancestors. Take your questions into your dreams — answers may come... sometimes takes time over three nights and days. Write down what you receive, for the messages may guide your steps into the light. |
||||
|
||||
== The Second Rauhnacht |
||||
|
||||
December 25th to 26th |
||||
|
||||
image::https://i.nostr.build/y0DXJuaASG4OhDPF.jpg[2nd Rauchnacht, 300] |
||||
|
||||
.The second mystical night of twelve |
||||
[%hardbreaks] |
||||
Month: *February* |
||||
Namessake: *St. Stephen* |
||||
Theme: *guidance and spiritual connection* |
||||
|
||||
The second Rauhnacht connects to the month of February and resonates with the energy of St. Stephen, symbolizing guidance and spiritual connection. |
||||
|
||||
This night invites you to return to your spiritual home, a sacred space within, where you are connected to the Source. Strengthen your bond with your higher self, knowing you are supported and accompanied by beings of light. |
||||
|
||||
* Visit your inner sanctuary through meditation, stillness, or guided journeys to uncover what is essential for the times ahead. Who or what will accompany you into the new year? |
||||
* Reflect on guides such as masters, angels, spirit animals, plants, minerals, or elements that resonate with you. |
||||
* What still seeks resolution or healing? |
||||
* Consider the relationships you wish to nurture. |
||||
* Write down any insights or symbols you receive. |
||||
* You may also create a small medicine pouch filled with meaningful tokens. Lastly, ask for the clarity to hear your inner voice more distinctly as you move forward. |
||||
|
||||
If you desire support for this inner journey, please feel free to reach out. |
||||
|
||||
== The Third Rauhnacht |
||||
|
||||
December 26th to 27th |
||||
|
||||
image::https://i.nostr.build/WG1m4uymsIlwYtNd.jpg[3rd Rauchnacht, 300] |
||||
|
||||
.The third mystical night of twelve |
||||
[%hardbreaks] |
||||
Month: *March* |
||||
Theme: *God's grace and unconditional love* |
||||
Namesake: *St. John* |
||||
|
||||
The third Rauhnacht corresponds to the month of March and is associated with St. John (Johannes), symbolizing God's grace and unconditional love. |
||||
|
||||
This night invites you to open your heart and allow miracles into your life. In March, nature prepares for breakthroughs and renewal – just as you can release old burdens to make space for something new. |
||||
|
||||
Ask yourself: |
||||
|
||||
* Who was there for me during difficult times? Who brought love, joy, or wise advice into my life? |
||||
* To whom did I offer my friendship and affection? |
||||
* Are there relationships that drain my energy or feel out of balance? |
||||
* Which connections do I want to strengthen, and which can I loosen? |
||||
|
||||
Take this opportunity to resolve old matters and open your heart anew through forgiveness and release. |
||||
|
||||
Breathe deeply in and out, and imagine your heart opening like a flower, allowing unconditional love to flow into you. Only by accepting and loving yourself can you truly love others. Recognize your own light and inner beauty – this is the key to experiencing miracles. |
||||
|
||||
Today, take time to formulate your wishes and goals for the new year. Focus on what brings you deep joy, love, and passion, as these are signposts guiding you to your soul's purpose. Create a vision board or write and draw your dreams, lighting a candle for each wish. Finally, hand your heartfelt desires over to the universe with trust, and open yourself |
||||
to the miracles that await. |
||||
|
||||
== The Fourth Rauhnacht |
||||
|
||||
December 27th to 28th |
||||
|
||||
image::https://i.nostr.build/Xf6xIjIJOiN47CW9.jpg[4th Rauchnacht, 300] |
||||
|
||||
.The fourth mystical night of twelve |
||||
[%hardbreaks] |
||||
Month: *April* |
||||
Theme: *release and transformation and Innocent Children* |
||||
|
||||
The fourth Rauhnacht corresponds to the month of April and symbolizes release and transformation. It is the Day of the Innocent Children and reminds us that sometimes the old tries to suppress or even destroy the new – whether through past actions, experiences, or personal patterns. Today is a powerful time to clear the past and set the stage for a positive new year. |
||||
|
||||
* Take some time to write down everything from the past that has burdened you, such as arguments, bad news, or nightmares. |
||||
* On a new piece of paper, reframe all the negative experiences into positive ones. Rewrite your story in a way that serves you and supports your well-being. |
||||
* Burn the first paper with the negative experiences as a symbolic act of release. |
||||
* Use visualization to transform: Imagine a violet flame flowing through and around your negative experiences, purifying and harmonizing them according to the divine plan. Feel how the negative energy shifts into positive energy. Visualize joy and gratitude as you see this transformation unfold. Hold the vision of your new images and allow white-golden light to flood through them, blessing and energizing your new vision. |
||||
|
||||
Finally, light a candle for your guides of light, whether an angel, master, or other beings. This light presence will accompany you throughout the coming year. |
||||
|
||||
Let go, trust that harmony is restored, and know that your new path is set. |
||||
|
||||
== The Fifth Rauhnacht |
||||
|
||||
December 28th to 29th |
||||
|
||||
image::https://i.nostr.build/Aj9U3RTjii4s9u4f.jpg[5th Rauchnacht, 300] |
||||
|
||||
.The fifth mystical night of twelve |
||||
[%hardbreaks] |
||||
Month: *May* |
||||
Theme: *Friendship* |
||||
Namesake: *Thomas* |
||||
|
||||
The fifth Rauhnacht corresponds to the month of May and honors the feast day of Thomas, meaning "Twin." This night, therefore, is about the soul's twin. |
||||
|
||||
The theme of this night is friendship – with yourself and with others. True friendships hold the quiet secret of deep connection. Take time to ask yourself: Am I my own best friend? Forgive yourself and be kind to yourself. Only by offering love and understanding to yourself can you extend it to others. |
||||
|
||||
* Reflect on the friendships in your life. Which ones enrich you? Are there broken connections that need healing? What would it take to find peace and resolution? Clearing the field can create space for new, meaningful friendships and experiences of love. |
||||
* Look back at the past year: What have you done well? Where were the highs and lows, the setbacks and successes? Speak to yourself as you would to your best friend, and say "Yes" to yourself. |
||||
* If there are people you wish to release energetically, visualize the infinity symbol. Imagine yourself standing in one circle and the other person in the opposite circle. See their light and become aware of all the connections formed between you over time. Ask a light being, an angel, or divine light to gently dissolve any heavy or unhealthy bonds. Visualise that both of you are wrapped in light as the infinity symbol softly separates in the center. The circles move apart until the distance feels light, comfortable, and harmonious for both. Thank the person for the love and shared experiences. Everything comes to a good end. |
||||
* Perhaps you’d like to tell your friends how much you love and appreciate them. This can be done through a letter, a phone call, or a heartfelt gesture. It's a beautiful way to celebrate connection and gratitude. |
||||
|
||||
== The Sixth Rauhnacht |
||||
|
||||
December 29th to 30th |
||||
|
||||
image::https://i.nostr.build/vvqa4EDyGh5O48Ab.png[6th Rauchnacht, 300] |
||||
|
||||
.The sixth mystical night of twelve |
||||
[%hardbreaks] |
||||
Month: *June* |
||||
Theme: *Purification* |
||||
|
||||
The sixth Rauhnacht is dedicated to the month of June and the theme of purification. This night is about letting go and looking ahead with confidence — expect the best! What do you want to leave behind from the old year? Take some time to reflect on the past year and consciously prepare for the transition into the new one. Today is also a beautiful opportunity to connect with your family. |
||||
|
||||
* Begin by reaching out to the soul of your mother and asking her for a gift to take into the new year. Form your hands into a bowl and receive this gift with gratitude. |
||||
* Then, connect with the soul of your father and do the same. These gifts come from the highest realm of love. |
||||
* If you have a partner, ask their soul for a gift and, in return, offer them one from your own highest love. |
||||
* If you have children, bestow gifts upon them from this sacred space as well. These gifts will accompany you and your family through the year ahead. |
||||
* Find a quiet and safe space to connect with your spirit. Reflect on the past year: |
||||
** What went well, and what did not? |
||||
** What do you wish to leave behind? |
||||
** Are there matters that still need cleansing or healing? |
||||
** What do you need to release fully so it no longer weighs you down? |
||||
* Write down your thoughts and burn the paper in a flame, ideally with white sage or incense (Olibanum), as a symbolic act of letting go. |
||||
* Additionally, clear your home of anything broken, unnecessary, or associated with negative feelings. By doing so, you create space for new energy and opportunities to enter your life. |
||||
|
||||
== The Seventh Rauhnacht |
||||
|
||||
December 30th to 31st |
||||
|
||||
image::https://i.nostr.build/vZjzmGs2eTxUxA3g.jpg[7th Rauchnacht, 300] |
||||
|
||||
.The seventh mystical night of twelve |
||||
[%hardbreaks] |
||||
Month: *July* |
||||
Theme: *Preparation for what’s to come* |
||||
Namesake: *Silvester* |
||||
|
||||
The seventh Rauhnacht, associated with the month of July and the namesake _Silvester_, carries the theme of _Preparation for What’s to Come._ |
||||
|
||||
It represents the gateway — the transition from a past phase into a new one. Since the introduction of the Gregorian calendar, this transition has been celebrated by many on December 31st, a day dedicated to the Roman bishop Silvester. His death commemorates the end of Christian persecution and the establishment of Christianity as a state religion. (For spiritual insights on Christ consciousness, I recommend https://njump.me/naddr1qvzqqqr4gupzp35mw8w9vn7ux59vmhle98e96usz4s28pjr53psgh4ke3epxhfmrqyvhwumn8ghj7un9d3shjtnndehhyapwwdhkx6tpdshszythwden5te0dehhxarj9emkjmn99uqp2d3ctaynzefcvex5vamxvf04sunzfu6nqdgtaz38t[this article I recently shared]). |
||||
|
||||
Every transition holds the opportunity to change, reshape, and perceive life with fresh eyes. Use this day to prepare for the new year: |
||||
|
||||
* Symbolically wash away the old — whether through bathing, showering, or a cleansing ritual—and visualize your wishes and dreams for the year ahead. |
||||
* Celebrate the transition in the circle of your loved ones, as the circle symbolizes unity and wholeness. |
||||
* This night is also about banishing unwanted energies. Cleanse your space with smudging tools such as sage or frankincense to dispel negativity. *Important: Smudge before midnight to avoid sweeping away the good fortune of the new year.* |
||||
* Set new, meaningful intentions and reflect on the past year: What have you accomplished, and what remains unresolved? |
||||
* Share a plate of food with the nature spirits by offering it at the roots of a tree to show your gratitude and strengthen their goodwill. |
||||
* New Year’s Eve is also a perfect time for oracle rituals, such as drawing cards, lead pouring, or pendulum readings. |
||||
* Wish your loved ones a peaceful, joyful, and healthy transition into the new year. |
||||
|
||||
This is also an ideal time to nurture your health and well-being. Rather than losing yourself in the hustle of crowds or indulging in excess, choose a more mindful approach — it will give you a strong and joyful start to the year. |
||||
|
||||
Fireworks were created to distract us from our inner processes. Smudging, on the other hand, brings you back to the present moment, centers you, and deepens your connection to your soul. |
||||
|
||||
#Much love to all of you, and heartfelt thanks for being together in this! It truly means so much to me.# |
||||
|
||||
== The Eighth Rauhnacht |
||||
|
||||
December 31st to January 1st |
||||
|
||||
image::https://i.nostr.build/2zCdzjOFRGb8ASLs.jpg[8th Rauchnacht, 300] |
||||
|
||||
.The eighth mystical night of twelve |
||||
[%hardbreaks] |
||||
Month: *August* |
||||
Theme: *Birth of the New Year* |
||||
|
||||
#Good fortune and blessings on all your paths.# |
||||
|
||||
This eighth Rauhnacht is dedicated to new beginnings and blessings. It is a time to welcome the New Year with celebration and set it on a path filled with happiness and positivity. |
||||
|
||||
* Bless the New Year by lighting candles — the glow invites good fortune and symbolizes the start of a fresh, fulfilling cycle. If you feel the need, you can also use incense to cleanse and harmonize the energy in your home. |
||||
* Remember: How you begin today will accompany you throughout the entire year. Make this day meaningful, whether through gratitude, loving connections, or powerful intentions for the year ahead. |
||||
* The dreams that visited you on New Year's Eve carry a special message. Take time to reflect on these dreams and write them down — they may hold guidance for the months to come. |
||||
|
||||
#Wishing you all the best for this new year filled with wonders and opportunities! May it be blessed with happiness, love, and endless possibilities.# |
||||
|
||||
== The Ninth Rauhnacht |
||||
|
||||
January 1st to January 2nd |
||||
|
||||
image::https://i.nostr.build/9YSicvZQYFdLIO7i.jpg[9th Rauchnacht, 300] |
||||
|
||||
.The ninth mystical night of twelve |
||||
[%hardbreaks] |
||||
Month: *September* |
||||
Theme: *Gold, Blessing Light* |
||||
Rune of the Day: *Kenaz* |
||||
Namesakes: *Saint Katharina and Kaspar* |
||||
|
||||
For many, January 2nd marks the return of everyday life. |
||||
|
||||
Saint Katharina represents various aspects of the Earth Mother. Her symbol, the Wheel of Life, embodies the cycle of birth, life, and death. She brings the light and blesses the Sun Child with the gifts of life and wisdom. The wheel also symbolizes the belief that through faith, destiny and redemption can be achieved. |
||||
|
||||
The name Kaspar means "treasurer" in Persian. He symbolizes gold, which he offers as a gift. Gold represents the perfection of the soul and the radiant purity of inner light. It lies at the center of the Wheel of Destiny, signifying harmony and balance. The light has been embodied, and it is perfect and complete. |
||||
|
||||
image::https://i.nostr.build/V8Jfre0erMhpft7A.jpg[Rune Kenaz, 150,, role=right] |
||||
The Rune Kenaz, the sixth rune, is interpreted as a torch or flame. It symbolizes insight, wisdom, and inner fire. This rune embodies the power of transformation through light and knowledge. It reminds us that we are all bearers of light – this light resides in every cell and every atom of our being. |
||||
|
||||
It is time to focus on your center and align yourself. In the center lies stillness, peace, and strength, from which all energy flows. Connect with the golden core within you. |
||||
|
||||
.A small blessing meditation for the Ninth Rauhnacht |
||||
Use this night to connect with your inner light and golden core: |
||||
|
||||
* _Light a Candle:_ The flame of the candle symbolizes a new beginning. |
||||
* _Sit in Stillness:_ Invite this light into your heart. |
||||
* _Visualize Golden Light:_ See this light flowing through your crown chakra into your heart and then radiating out through your hands. |
||||
* _Bless the New Year:_ Guide this golden stream of light through the 12 months of the year. Wrap each month in golden light, blessing it with love and intention. |
||||
* _Notice Your Sensations:_ Pay attention to the energy flow and your feelings during this meditation. Reflect on how each month feels as the light touches it. |
||||
* _Focus on Your Center:_ Reconnect with the inner peace and harmony within your core. From this place of stillness, let your strength for the year ahead grow. |
||||
|
||||
May this night guide you to your light and remind you that the gold of perfection and wisdom already resides within you. Bless the new year with light, love, and the power of your inner flame. |
||||
|
||||
== Tenth Rauhnacht |
||||
|
||||
January 2nd to January 3rd |
||||
|
||||
image::https://i.nostr.build/MjMet1uDZ6bVbjag.jpg[10th Rauchnacht, 300] |
||||
|
||||
.The tenth mystical night of twelve |
||||
[%hardbreaks] |
||||
Month: *October* |
||||
Theme: *Frankincense, Visions and Insights, Connection to the Divine* |
||||
Namesakes: *Ehwaz, Ambeth-Margarete, Melchior* |
||||
|
||||
January 3rd invites us to connect with our inner wisdom and divine inspiration. |
||||
|
||||
image::https://i.nostr.build/46hv92Ay7WHJ4Wcr.jpg[Rune Ahwaz, 150,, role=left] |
||||
The Rune Ahwaz (or Ehwaz) stands for M and represents movement, development, and progress. It |
||||
reminds us that life is in constant flux, and through transformation, we grow. |
||||
|
||||
Ambeth-Margarete embodies the nurturing mother, fertility, abundance, birth, and rebirth – filled with light, love, and goodness. She represents the spiral of life and the cauldron of abundance, offering us tireless strength and energy. |
||||
|
||||
Melchior, the wise man from the East, brings frankincense to the crib. Frankincense symbolizes the life’s gold within us and our connection to the Divine. |
||||
|
||||
This night calls us to reflect on our gifts, imagination, and life energy: |
||||
|
||||
* How do I use my talents, thoughts, and creativity? |
||||
* How do I invest my life energy – in words, actions, and expression? |
||||
* What occupies most of my time? |
||||
|
||||
Each day provides a new opportunity to realign yourself. Reflect: |
||||
|
||||
* Which aspects of your life do you want to transform and improve? |
||||
* What changes do you wish to make this year? What steps can you take to bring them to life? |
||||
* Visualize your future: Create clear images in your mind of how you want to shape your life. |
||||
|
||||
#Every transformation begins within – it's in your hands.# |
||||
|
||||
The only constant is change. Use the energy of this day to explore new perspectives and |
||||
align yourself with your vision for the months ahead. |
||||
|
||||
== Eleventh Rauhnacht |
||||
|
||||
January 3rd to January 4th |
||||
|
||||
image::https://i.nostr.build/5MOsBHj4N3abJyWK.webp[11th Rauchnacht, 300] |
||||
|
||||
.The eleventh mystical night of twelve |
||||
[%hardbreaks] |
||||
Month: *November* |
||||
Theme: *Letting Go, Farewells, Reflection on Death* |
||||
Namesake: *Berkana, Borbeth-Barbara, Balthasar* |
||||
|
||||
This night invites us to pause and reflect on impermanence and the act of letting go. |
||||
|
||||
image::https://i.nostr.build/e4eKs1vwkkxQ2cuc.webp[Rune Berkana, 150,, role=right] |
||||
The rune Berkana symbolizes fertility and the creative force of Mother Earth. It represents both birth and death, promising a peaceful and fruitful life in harmony with nature’s cycles. Borbeth-Barbara, whose name derives from the Earth goddess Borbeth, stands for severing life’s threads and releasing old ties. Her symbol, the tower, reflects strength and transition. The name "Borbeth" evokes words such as bed — a place of rest — prayer, and petition. Barbara signifies both the act of giving life and the bier upon which the deceased rest. |
||||
|
||||
Balthasar, the wise man from the East, means _God protect his eternal life._ He brings myrrh, which was traditionally used for embalming the dead. Myrrh represents the transient nature of human life, reminding us that our time on Earth is finite. |
||||
|
||||
Reflecting on life’s impermanence leads us to profound existential questions: |
||||
|
||||
* Why am I here? |
||||
* What do I want to bring into this world? |
||||
* What is the purpose of my life? |
||||
* What goal am I pursuing during my time on Earth? |
||||
|
||||
Center yourself to find stillness and clarity for realignment. Take time to think about death and the questions it raises. Death isn’t only the end of life but also manifests daily in goodbyes, transitions, and transformations. |
||||
|
||||
Ask yourself: |
||||
|
||||
* What has truly ended, and what needs to be let go of? |
||||
* Where do I feel trapped, and what do I want to free myself from? |
||||
* What is truly important to me, and how do I want to use my life energy? |
||||
* What legacy do I want to leave behind? What do I want to be said or written about me? |
||||
|
||||
This night offers the chance to bring order to life, release old ties, and refocus on what truly matters. By consciously engaging with the act of letting go, you create space for new beginnings — laying the foundation for a meaningful and fulfilling life path. |
||||
|
||||
== The Twelfth Rauhnacht |
||||
|
||||
January 4th to January 5th |
||||
|
||||
image::https://i.nostr.build/FcdIK0RWD3dzYI3N.jpg[12th Rauchnacht, 300] |
||||
|
||||
.The twelfth mystical night of twelve |
||||
[%hardbreaks] |
||||
Month: *December* |
||||
Theme: *Cleansing and Purification* |
||||
Namesake: *The Three Holy Maidens – Day of Grace* |
||||
|
||||
The twelfth Rauhnacht marks a time of closure and cleansing. The Three Holy Maidens, representing the sacred feminine in its various aspects, stand for grace, healing, and protection. They guide us during this night with their blessing, showing the path to inner and outer harmony. |
||||
|
||||
Tonight offers you the opportunity to address and release anything that may not have gone well during the past 12 nights. |
||||
|
||||
Take some time to reflect on the Rauhnächte: |
||||
|
||||
* What has weighed on you? |
||||
* Which signs or experiences felt unsettling or uncomfortable? |
||||
|
||||
This night is filled with unique power and energy. With focused reflection, you can align yourself with this special time and prepare for renewal. |
||||
|
||||
In many places, Percht Processions are held, and old customs and traditions are celebrated. For example, the Three Kings Cake is traditionally baked on January 6th. |
||||
|
||||
Another practice involves smudging hats, magical items, ritual tools, and healing stones to clear away old and burdensome energies. |
||||
|
||||
During this night, strong forces are believed to converge, making it advisable not to spend too much time outdoors. It is said that dreams from this night have the power to come true. |
||||
|
||||
.A ritual for this night |
||||
Before going to bed, smudge everything that is important to you and has been exposed to various energies: |
||||
|
||||
* Your home’s entrance or door |
||||
* Your coat or shoes |
||||
* Your bedroom or other spaces |
||||
|
||||
Go to sleep with the clear intention of feeling protected by beings of light as you journey through your dreamland. |
||||
|
||||
This final Rauhnacht closes the cycle and prepares the way for the coming year — cleansed, blessed, and filled with positive energy. |
||||
|
||||
#I hope you enjoyed this journey of the 12 Rauhnächte. I'd love to hear your opinion and reflections.# |
||||
File diff suppressed because it is too large
Load Diff
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@@ -0,0 +1,395 @@
|
||||
= Bitcoin: A Peer-to-Peer Electronic Cash System |
||||
|
||||
== Abstract |
||||
|
||||
A purely peer-to-peer version of electronic cash would allow online payments to be sent directly from one party to another without going through a financial institution. Digital signatures provide part of the solution, but the main benefits are lost if a trusted third party is still required to prevent double-spending. We propose a solution to the double-spending problem using a peer-to-peer network. The network timestamps transactions by hashing them into an ongoing chain of hash-based proof-of-work, forming a record that cannot be changed without redoing the proof-of-work. The longest chain not only serves as proof of the sequence of events witnessed, but proof that it came from the largest pool of CPU power. As long as a majority of CPU power is controlled by nodes that are not cooperating to attack the network, they'll generate the longest chain and outpace attackers. The network itself requires minimal structure. Messages are broadcast on a best effort basis, and nodes can leave and rejoin the network at will, accepting the longest proof-of-work chain as proof of what happened while they were gone. |
||||
|
||||
== 1. Introduction |
||||
|
||||
Commerce on the Internet has come to rely almost exclusively on financial institutions serving as trusted third parties to process electronic payments. While the system works well enough for most transactions, it still suffers from the inherent weaknesses of the trust based model. Completely non-reversible transactions are not really possible, since financial institutions cannot avoid mediating disputes. The cost of mediation increases transaction costs, limiting the minimum practical transaction size and cutting off the possibility for small casual transactions, and there is a broader cost in the loss of ability to make non-reversible payments for non-reversible services. With the possibility of reversal, the need for trust spreads. Merchants must be wary of their customers, hassling them for more information than they would otherwise need. A certain percentage of fraud is accepted as unavoidable. These costs and payment uncertainties can be avoided in person by using physical currency, but no mechanism exists to make payments over a communications channel without a trusted party. |
||||
|
||||
What is needed is an electronic payment system based on cryptographic proof instead of trust, allowing any two willing parties to transact directly with each other without the need for a trusted third party. Transactions that are computationally impractical to reverse would protect sellers from fraud, and routine escrow mechanisms could easily be implemented to protect buyers. In this paper, we propose a solution to the double-spending problem using a peer-to-peer distributed timestamp server to generate computational proof of the chronological order of transactions. The system is secure as long as honest nodes collectively control more CPU power than any cooperating group of attacker nodes. |
||||
|
||||
== 2. Transactions |
||||
|
||||
We define an electronic coin as a chain of digital signatures. Each owner transfers the coin to the next by digitally signing a hash of the previous transaction and the public key of the next owner and adding these to the end of the coin. A payee can verify the signatures to verify the chain of ownership. |
||||
|
||||
[source] |
||||
---- |
||||
┌─────────────────────┐ ┌─────────────────────┐ ┌─────────────────────┐ |
||||
│ │ │ │ │ │ |
||||
│ Transaction │ │ Transaction │ │ Transaction │ |
||||
│ │ │ │ │ │ |
||||
│ ┌─────────────┐ │ │ ┌─────────────┐ │ │ ┌─────────────┐ │ |
||||
│ │ Owner 1's │ │ │ │ Owner 2's │ │ │ │ Owner 3's │ │ |
||||
│ │ Public Key │ │ │ │ Public Key │ │ │ │ Public Key │ │ |
||||
│ └───────┬─────┘ │ │ └───────┬─────┘ │ │ └───────┬─────┘ │ |
||||
│ │ . │ │ │ . │ │ │ │ |
||||
──────┼─────────┐ │ . ├───────────────┼─────────┐ │ . ├──────────────┼─────────┐ │ │ |
||||
│ │ │ . │ │ │ │ . │ │ │ │ │ |
||||
│ ┌──▼─▼──┐ . │ │ ┌──▼─▼──┐ . │ │ ┌──▼─▼──┐ │ |
||||
│ │ Hash │ . │ │ │ Hash │ . │ │ │ Hash │ │ |
||||
│ └───┬───┘ . │ Verify │ └───┬───┘ . │ Verify │ └───┬───┘ │ |
||||
│ │ ............................ │ ........................... │ │ |
||||
│ │ │ │ │ │ │ │ │ │ │ |
||||
│ ┌──────▼──────┐ │ │ ┌─▼────▼──────┐ │ │ ┌─▼────▼──────┐ │ |
||||
│ │ Owner 0's │ │ Sign │ │ Owner 1's │ │ Sign │ │ Owner 2's │ │ |
||||
│ │ Signature │ │ ...........─►│ Signature │ │ ...........─►│ Signature │ │ |
||||
│ └─────────────┘ │ . │ └─────────────┘ │ . │ └─────────────┘ │ |
||||
│ │ . │ │ . │ │ |
||||
└─────────────────────┘ . └─────────────────────┘ . └─────────────────────┘ |
||||
. . |
||||
┌─────────────┐ . ┌─────────────┐ . ┌─────────────┐ |
||||
│ Owner 1's │........... │ Owner 2's │.......... │ Owner 3's │ |
||||
│ Private Key │ │ Private Key │ │ Private Key │ |
||||
└─────────────┘ └─────────────┘ └─────────────┘ |
||||
---- |
||||
|
||||
The problem of course is the payee can't verify that one of the owners did not double-spend the coin. A common solution is to introduce a trusted central authority, or mint, that checks every transaction for double spending. After each transaction, the coin must be returned to the mint to issue a new coin, and only coins issued directly from the mint are trusted not to be double-spent. The problem with this solution is that the fate of the entire money system depends on the company running the mint, with every transaction having to go through them, just like a bank. |
||||
|
||||
We need a way for the payee to know that the previous owners did not sign any earlier transactions. For our purposes, the earliest transaction is the one that counts, so we don't care about later attempts to double-spend. The only way to confirm the absence of a transaction is to be aware of all transactions. In the mint based model, the mint was aware of all transactions and decided which arrived first. To accomplish this without a trusted party, transactions must be publicly announcedfootnote:B-Money[W. Dai, "b-money," http://www.weidai.com/bmoney.txt, 1998.], and we need a system for participants to agree on a single history of the order in which they were received. The payee needs proof that at the time of each transaction, the majority of nodes agreed it was the first received. |
||||
|
||||
== 3. Timestamp Server |
||||
|
||||
The solution we propose begins with a timestamp server. A timestamp server works by taking a hash of a block of items to be timestamped and widely publishing the hash, such as in a newspaper or Usenet postfootnote:Design[H. Massias, X.S. Avila, and J.-J. Quisquater, "Design of a secure timestamping service with minimal |
||||
trust requirements," In 20th Symposium on Information Theory in the Benelux, May 1999.]footnote:How-to[S. Haber, W.S. Stornetta, "How to time-stamp a digital document," In Journal of Cryptology, vol 3, no 2, pages 99-111, 1991.]footnote:Improving[D. Bayer, S. Haber, W.S. Stornetta, "Improving the efficiency and reliability of digital time-stamping", In Sequences II: Methods in Communication, Security and Computer Science, pages 329-334, 1993.]footnote:Secure[S. Haber, W.S. Stornetta, "Secure names for bit-strings," In Proceedings of the 4th ACM Conference on Computer and Communications Security, pages 28-35, April 1997.]. The timestamp proves that the data must have existed at the time, obviously, in order to get into the hash. Each timestamp includes the previous timestamp in its hash, forming a chain, with each additional timestamp reinforcing the ones before it. |
||||
|
||||
[source] |
||||
---- |
||||
┌──────┐ ┌──────┐ |
||||
────────────►│ ├───────────────────────►│ ├───────────────────► |
||||
│ Hash │ │ Hash │ |
||||
┌───►│ │ ┌───►│ │ |
||||
│ └──────┘ │ └──────┘ |
||||
│ │ |
||||
┌┴──────────────────────────┐ ┌┴──────────────────────────┐ |
||||
│ Block │ │ Block │ |
||||
│ ┌─────┐ ┌─────┐ ┌─────┐ │ │ ┌─────┐ ┌─────┐ ┌─────┐ │ |
||||
│ │Item │ │Item │ │... │ │ │ │Item │ │Item │ │... │ │ |
||||
│ └─────┘ └─────┘ └─────┘ │ │ └─────┘ └─────┘ └─────┘ │ |
||||
│ │ │ │ |
||||
└───────────────────────────┘ └───────────────────────────┘ |
||||
---- |
||||
|
||||
== 4. Proof-of-Work |
||||
|
||||
To implement a distributed timestamp server on a peer-to-peer basis, we will need to use a proof-of-work system similar to Adam Back's Hashcashfootnote:Hashcash[A. Back, "Hashcash - a denial of service counter-measure", http://www.hashcash.org/papers/hashcash.pdf, 2002.], rather than newspaper or Usenet posts. The proof-of-work involves scanning for a value that when hashed, such as with SHA-256, the hash begins with a number of zero bits. The average work required is exponential in the number of zero bits required and can be verified by executing a single hash. |
||||
|
||||
For our timestamp network, we implement the proof-of-work by incrementing a nonce in the block until a value is found that gives the block's hash the required zero bits. Once the CPU effort has been expended to make it satisfy the proof-of-work, the block cannot be changed without redoing the work. As later blocks are chained after it, the work to change the block would include redoing all the blocks after it. |
||||
|
||||
[source] |
||||
---- |
||||
┌────────────────────────────────────────┐ ┌────────────────────────────────────────┐ |
||||
│ Block │ │ Block │ |
||||
│ ┌──────────────────┐ ┌──────────────┐ │ │ ┌──────────────────┐ ┌──────────────┐ │ |
||||
───────┼─►│ Prev Hash │ │ Nonce │ ├──────┼─►│ Prev Hash │ │ Nonce │ │ |
||||
│ └──────────────────┘ └──────────────┘ │ │ └──────────────────┘ └──────────────┘ │ |
||||
│ │ │ │ |
||||
│ ┌──────────┐ ┌──────────┐ ┌──────────┐ │ │ ┌──────────┐ ┌──────────┐ ┌──────────┐ │ |
||||
│ │ Tx │ │ Tx │ │ ... │ │ │ │ Tx │ │ Tx │ │ ... │ │ |
||||
│ └──────────┘ └──────────┘ └──────────┘ │ │ └──────────┘ └──────────┘ └──────────┘ │ |
||||
│ │ │ │ |
||||
└────────────────────────────────────────┘ └────────────────────────────────────────┘ |
||||
---- |
||||
|
||||
The proof-of-work also solves the problem of determining representation in majority decision making. If the majority were based on one-IP-address-one-vote, it could be subverted by anyone able to allocate many IPs. Proof-of-work is essentially one-CPU-one-vote. The majority decision is represented by the longest chain, which has the greatest proof-of-work effort invested in it. If a majority of CPU power is controlled by honest nodes, the honest chain will grow the fastest and outpace any competing chains. To modify a past block, an attacker would have to redo the proof-of-work of the block and all blocks after it and then catch up with and surpass the work of the honest nodes. We will show later that the probability of a slower attacker catching up diminishes exponentially as subsequent blocks are added. |
||||
|
||||
To compensate for increasing hardware speed and varying interest in running nodes over time, the proof-of-work difficulty is determined by a moving average targeting an average number of blocks per hour. If they're generated too fast, the difficulty increases. |
||||
|
||||
== 5. Network |
||||
|
||||
The steps to run the network are as follows: |
||||
|
||||
. New transactions are broadcast to all nodes. |
||||
. Each node collects new transactions into a block. |
||||
. Each node works on finding a difficult proof-of-work for its block. |
||||
. When a node finds a proof-of-work, it broadcasts the block to all nodes. |
||||
. Nodes accept the block only if all transactions in it are valid and not already spent. |
||||
. Nodes express their acceptance of the block by working on creating the next block in the chain, using the hash of the accepted block as the previous hash. |
||||
|
||||
Nodes always consider the longest chain to be the correct one and will keep working on extending it. If two nodes broadcast different versions of the next block simultaneously, some nodes may receive one or the other first. In that case, they work on the first one they received, but save the other branch in case it becomes longer. The tie will be broken when the next proof-of-work is found and one branch becomes longer; the nodes that were working on the other branch will then switch to the longer one. |
||||
|
||||
New transaction broadcasts do not necessarily need to reach all nodes. As long as they reach many nodes, they will get into a block before long. Block broadcasts are also tolerant of dropped messages. If a node does not receive a block, it will request it when it receives the next block and realizes it missed one. |
||||
|
||||
== 6. Incentive |
||||
|
||||
By convention, the first transaction in a block is a special transaction that starts a new coin owned by the creator of the block. This adds an incentive for nodes to support the network, and provides a way to initially distribute coins into circulation, since there is no central authority to issue them. The steady addition of a constant of amount of new coins is analogous to gold miners expending resources to add gold to circulation. In our case, it is CPU time and electricity that is expended. |
||||
|
||||
The incentive can also be funded with transaction fees. If the output value of a transaction is less than its input value, the difference is a transaction fee that is added to the incentive value of the block containing the transaction. Once a predetermined number of coins have entered circulation, the incentive can transition entirely to transaction fees and be completely inflation free. |
||||
|
||||
The incentive may help encourage nodes to stay honest. If a greedy attacker is able to assemble more CPU power than all the honest nodes, he would have to choose between using it to defraud people by stealing back his payments, or using it to generate new coins. He ought to find it more profitable to play by the rules, such rules that favour him with more new coins than everyone else combined, than to undermine the system and the validity of his own wealth. |
||||
|
||||
== 7. Reclaiming Disk Space |
||||
|
||||
Once the latest transaction in a coin is buried under enough blocks, the spent transactions before it can be discarded to save disk space. To facilitate this without breaking the block's hash, transactions are hashed in a Merkle Treefootnote:Protocols[R.C. Merkle, "Protocols for public key cryptosystems," In Proc. 1980 Symposium on Security and Privacy, IEEE Computer Society, pages 122-133, April 1980.]footnote:Design[H. Massias, X.S. Avila, and J.-J. Quisquater, "Design of a secure timestamping service with minimal trust requirements," In 20th Symposium on Information Theory in the Benelux, May 1999.]footnote:Secure[S. Haber, W.S. Stornetta, "Secure names for bit-strings," In Proceedings of the 4th ACM Conference |
||||
on Computer and Communications Security, pages 28-35, April 1997.], with only the root included in the block's hash. Old blocks can then be compacted by stubbing off branches of the tree. The interior hashes do not need to be stored. |
||||
|
||||
[source] |
||||
---- |
||||
┌──────────────────────────────────────────┐ ┌──────────────────────────────────────────┐ |
||||
│ │ │ │ |
||||
│ Block ┌─────────────────────────────┐ │ │ Block ┌─────────────────────────────┐ │ |
||||
│ │ Block Header (Block Hash) │ │ │ │ Block Header (Block Hash) │ │ |
||||
│ │ ┌────────────┐ ┌─────────┐ │ │ │ │ ┌────────────┐ ┌─────────┐ │ │ |
||||
│ │ │ Prev Hash │ │ Nonce │ │ │ │ │ │ Prev Hash │ │ Nonce │ │ │ |
||||
│ │ └────────────┘ └─────────┘ │ │ │ │ └────────────┘ └─────────┘ │ │ |
||||
│ │ │ │ │ │ │ │ |
||||
│ │ ┌─────────────┐ │ │ │ │ ┌─────────────┐ │ │ |
||||
│ │ │ Root Hash │ │ │ │ │ │ Root Hash │ │ │ |
||||
│ │ └─────▲─▲─────┘ │ │ │ │ └─────▲─▲─────┘ │ │ |
||||
│ │ │ │ │ │ │ │ │ │ │ │ |
||||
│ │ │ │ │ │ │ │ │ │ │ │ |
||||
│ └───────────┼─┼───────────────┘ │ │ └───────────┼─┼───────────────┘ │ |
||||
│ │ │ │ │ │ │ │ |
||||
│ .......... │ │ .......... │ │ ┌────────┐ │ │ .......... │ |
||||
│ . ─────┘ └─────. . │ │ │ ├────┘ └─────. . │ |
||||
│ . Hash01 . . Hash23 . │ │ │ Hash01 │ . Hash23 . │ |
||||
│ .▲.....▲.. .▲.....▲.. │ │ │ │ .▲.....▲.. │ |
||||
│ │ │ │ │ │ │ └────────┘ │ │ │ |
||||
│ │ │ │ │ │ │ │ │ │ |
||||
│ │ │ │ │ │ │ │ │ │ |
||||
│ .....│.. ..│..... .....│.. ..│..... │ │ ┌────┴─┐ ..│..... │ |
||||
│ . . . . . . . . │ │ │ │ . . │ |
||||
│ .Hash0 . .Hash1 . .Hash2 . .Hash3 . │ │ │Hash2 │ .Hash3 . │ |
||||
│ ...▲.... ...▲.... ...▲.... ...▲.... │ │ │ │ . . │ |
||||
│ │ │ │ │ │ │ └──────┘ ...▲.... │ |
||||
│ │ │ │ │ │ │ │ │ |
||||
│ │ │ │ │ │ │ │ │ |
||||
│ ┌──┴───┐ ┌──┴───┐ ┌──┴───┐ ┌──┴───┐ │ │ ┌──┴───┐ │ |
||||
│ │ Tx0 │ │ Tx1 │ │ Tx2 │ │ Tx3 │ │ │ │ Tx3 │ │ |
||||
│ └──────┘ └──────┘ └──────┘ └──────┘ │ │ └──────┘ │ |
||||
│ │ │ │ |
||||
└──────────────────────────────────────────┘ └──────────────────────────────────────────┘ |
||||
Transactions Hashed in a Merkle Tree After Pruning Tx0-2 from the Block |
||||
---- |
||||
|
||||
A block header with no transactions would be about 80 bytes. If we suppose blocks are generated every 10 minutes, 80 bytes * 6 * 24 * 365 = 4.2MB per year. With computer systems typically selling with 2GB of RAM as of 2008, and Moore's Law predicting current growth of 1.2GB per year, storage should not be a problem even if the block headers must be kept in memory. |
||||
|
||||
== 8. Simplified Payment Verification |
||||
|
||||
It is possible to verify payments without running a full network node. A user only needs to keep a copy of the block headers of the longest proof-of-work chain, which he can get by querying network nodes until he's convinced he has the longest chain, and obtain the Merkle branch linking the transaction to the block it's timestamped in. He can't check the transaction for himself, but by linking it to a place in the chain, he can see that a network node has accepted it, and blocks added after it further confirm the network has accepted it. |
||||
|
||||
[source] |
||||
---- |
||||
Longest Proof-of-Work Chain |
||||
┌────────────────────────────────────────┐ ┌────────────────────────────────────────┐ ┌────────────────────────────────────────┐ |
||||
│ Block Header │ │ Block Header │ │ Block Header │ |
||||
│ ┌──────────────────┐ ┌──────────────┐ │ │ ┌──────────────────┐ ┌──────────────┐ │ │ ┌──────────────────┐ ┌──────────────┐ │ |
||||
───────┼─►│ Prev Hash │ │ Nonce │ ├──────┼─►│ Prev Hash │ │ Nonce │ ├───────┼─►│ Prev Hash │ │ Nonce │ ├────────► |
||||
│ └──────────────────┘ └──────────────┘ │ │ └──────────────────┘ └──────────────┘ │ │ └──────────────────┘ └──────────────┘ │ |
||||
│ │ │ │ │ │ |
||||
│ ┌───────────────────┐ │ │ ┌────────────────────┐ │ │ ┌───────────────────┐ │ |
||||
│ │ Merkle Root │ │ │ │ Merkle Root │ │ │ │ Merkle Root │ │ |
||||
│ └───────────────────┘ │ │ └────────▲─▲─────────┘ │ │ └───────────────────┘ │ |
||||
│ │ │ │ │ │ │ │ |
||||
└────────────────────────────────────────┘ └─────────────┼─┼────────────────────────┘ └────────────────────────────────────────┘ |
||||
│ │ |
||||
│ │ |
||||
┌────────┐ │ │ .......... |
||||
│ ├────┘ └─────. . |
||||
│ Hash01 │ . Hash23 . |
||||
│ │ .▲.....▲.. |
||||
└────────┘ │ │ |
||||
│ │ |
||||
│ │ Merkle Branch for Tx3 |
||||
│ │ |
||||
┌─────┴─┐ ..│..... |
||||
│ │ . . |
||||
│ Hash2 │ .Hash3 . |
||||
│ │ . . |
||||
└───────┘ ...▲.... |
||||
│ |
||||
│ |
||||
┌───┴───┐ |
||||
│ Tx3 │ |
||||
└───────┘ |
||||
---- |
||||
|
||||
As such, the verification is reliable as long as honest nodes control the network, but is more vulnerable if the network is overpowered by an attacker. While network nodes can verify transactions for themselves, the simplified method can be fooled by an attacker's fabricated transactions for as long as the attacker can continue to overpower the network. One strategy to protect against this would be to accept alerts from network nodes when they detect an invalid block, prompting the user's software to download the full block and alerted transactions to confirm the inconsistency. Businesses that receive frequent payments will probably still want to run their own nodes for more independent security and quicker verification. |
||||
|
||||
== 9. Combining and Splitting Value |
||||
|
||||
Although it would be possible to handle coins individually, it would be unwieldy to make a separate transaction for every cent in a transfer. To allow value to be split and combined, transactions contain multiple inputs and outputs. Normally there will be either a single input from a larger previous transaction or multiple inputs combining smaller amounts, and at most two outputs: one for the payment, and one returning the change, if any, back to the sender. |
||||
|
||||
[source] |
||||
---- |
||||
┌──────────────────────┐ |
||||
│ Transaction │ |
||||
│ │ |
||||
│ ┌─────┐ ┌─────┐ │ |
||||
─────┼──►│ in │ │ out │ ──┼─────► |
||||
│ └─────┘ └─────┘ │ |
||||
│ │ |
||||
│ │ |
||||
│ ┌─────┐ ┌─────┐ │ |
||||
─────┼──►│ in │ │ ... │ ──┼─────► |
||||
│ └─────┘ └─────┘ │ |
||||
│ │ |
||||
│ │ |
||||
│ ┌─────┐ │ |
||||
─────┼──►│... │ │ |
||||
│ └─────┘ │ |
||||
│ │ |
||||
└──────────────────────┘ |
||||
---- |
||||
|
||||
It should be noted that fan-out, where a transaction depends on several transactions, and those transactions depend on many more, is not a problem here. There is never the need to extract a complete standalone copy of a transaction's history. |
||||
|
||||
== 10. Privacy |
||||
|
||||
The traditional banking model achieves a level of privacy by limiting access to information to the parties involved and the trusted third party. The necessity to announce all transactions publicly precludes this method, but privacy can still be maintained by breaking the flow of information in another place: by keeping public keys anonymous. The public can see that someone is sending an amount to someone else, but without information linking the transaction to anyone. This is similar to the level of information released by stock exchanges, where the time and size of individual trades, the "tape", is made public, but without telling who the parties were. |
||||
|
||||
[source] |
||||
---- |
||||
Traditional Privacy Models │ |
||||
┌─────────────┐ ┌──────────────┐ │ ┌────────┐ |
||||
┌──────────────┐ ┌──────────────┐ │ Trusted │ │ │ │ │ │ |
||||
│ Identities ├──┤ Transactions ├───►│ Third Party ├──►│ Counterparty │ │ │ Public │ |
||||
└──────────────┘ └──────────────┘ │ │ │ │ │ │ │ |
||||
└─────────────┘ └──────────────┘ │ └────────┘ |
||||
│ |
||||
|
||||
New Privacy Model |
||||
┌────────┐ |
||||
┌──────────────┐ │ ┌──────────────┐ │ │ |
||||
│ Identities │ │ │ Transactions ├───►│ Public │ |
||||
└──────────────┘ │ └──────────────┘ │ │ |
||||
└────────┘ |
||||
---- |
||||
|
||||
As an additional firewall, a new key pair should be used for each transaction to keep them from being linked to a common owner. Some linking is still unavoidable with multi-input transactions, which necessarily reveal that their inputs were owned by the same owner. The risk is that if the owner of a key is revealed, linking could reveal other transactions that belonged to the same owner. |
||||
|
||||
== 11. Calculations |
||||
We consider the scenario of an attacker trying to generate an alternate chain faster than the honest chain. Even if this is accomplished, it does not throw the system open to arbitrary changes, such as creating value out of thin air or taking money that never belonged to the attacker. Nodes are not going to accept an invalid transaction as payment, and honest nodes will never accept a block containing them. An attacker can only try to change one of his own transactions to take back money he recently spent. |
||||
|
||||
The race between the honest chain and an attacker chain can be characterized as a Binomial Random Walk. The success event is the honest chain being extended by one block, increasing its lead by +1, and the failure event is the attacker's chain being extended by one block, reducing the gap by -1. |
||||
|
||||
The probability of an attacker catching up from a given deficit is analogous to a Gambler's Ruin problem. Suppose a gambler with unlimited credit starts at a deficit and plays potentially an infinite number of trials to try to reach breakeven. We can calculate the probability he ever reaches breakeven, or that an attacker ever catches up with the honest chain, as followsfootnote:Introduction[W. Feller, "An introduction to probability theory and its applications," 1957.]: |
||||
|
||||
[source] |
||||
---- |
||||
p = probability an honest node finds the next block< |
||||
q = probability the attacker finds the next block |
||||
q = probability the attacker will ever catch up from z blocks behind |
||||
---- |
||||
|
||||
[source] |
||||
---- |
||||
$$ |
||||
qz = |
||||
\begin{cases} |
||||
1 & \text{if } p \leq q \\ |
||||
\left(\frac{q}{p}\right) z & \text{if } p > q |
||||
\end{cases} |
||||
$$ |
||||
---- |
||||
|
||||
Given our assumption that p > q, the probability drops exponentially as the number of blocks the attacker has to catch up with increases. With the odds against him, if he doesn't make a lucky lunge forward early on, his chances become vanishingly small as he falls further behind. |
||||
|
||||
We now consider how long the recipient of a new transaction needs to wait before being sufficiently certain the sender can't change the transaction. We assume the sender is an attacker who wants to make the recipient believe he paid him for a while, then switch it to pay back to himself after some time has passed. The receiver will be alerted when that happens, but the sender hopes it will be too late. |
||||
|
||||
The receiver generates a new key pair and gives the public key to the sender shortly before signing. This prevents the sender from preparing a chain of blocks ahead of time by working on it continuously until he is lucky enough to get far enough ahead, then executing the transaction at that moment. Once the transaction is sent, the dishonest sender starts working in secret on a parallel chain containing an alternate version of his transaction. |
||||
|
||||
The recipient waits until the transaction has been added to a block and z blocks have been linked after it. He doesn't know the exact amount of progress the attacker has made, but assuming the honest blocks took the average expected time per block, the attacker's potential progress will be a Poisson distribution with expected value: |
||||
|
||||
[source] |
||||
---- |
||||
$$ |
||||
\lambda = z\frac{q}{p} |
||||
$$ |
||||
---- |
||||
|
||||
To get the probability the attacker could still catch up now, we multiply the Poisson density for each amount of progress he could have made by the probability he could catch up from that point: |
||||
|
||||
[source] |
||||
---- |
||||
$$ |
||||
\sum_{k=0}^{\infty} \frac{\lambda^k e^{-\lambda}}{k!} \cdot \left\{ |
||||
\begin{array}{cl} |
||||
\left(\frac{q}{p}\right)^{(z-k)} & \text{if } k \leq z \\ |
||||
1 & \text{if } k > z |
||||
\end{array} |
||||
\right. |
||||
$$ |
||||
---- |
||||
|
||||
Rearranging to avoid summing the infinite tail of the distribution... |
||||
|
||||
[source] |
||||
---- |
||||
$$ |
||||
1 - \sum_{k=0}^{z} \frac{\lambda^k e^{-\lambda}}{k!} \left(1-\left(\frac{q}{p}\right)^{(z-k)}\right) |
||||
$$ |
||||
---- |
||||
|
||||
Converting to C code... |
||||
|
||||
[source] |
||||
---- |
||||
#include <math.h> |
||||
|
||||
double AttackerSuccessProbability(double q, int z) |
||||
{ |
||||
double p = 1.0 - q; |
||||
double lambda = z * (q / p); |
||||
double sum = 1.0; |
||||
int i, k; |
||||
for (k = 0; k <= z; k++) |
||||
{ |
||||
double poisson = exp(-lambda); |
||||
for (i = 1; i <= k; i++) |
||||
poisson *= lambda / i; |
||||
sum -= poisson * (1 - pow(q / p, z - k)); |
||||
} |
||||
return sum; |
||||
} |
||||
---- |
||||
|
||||
Running some results, we can see the probability drop off exponentially with z. |
||||
|
||||
[source] |
||||
---- |
||||
q=0.1 |
||||
z=0 P=1.0000000 |
||||
z=1 P=0.2045873 |
||||
z=2 P=0.0509779 |
||||
z=3 P=0.0131722 |
||||
z=4 P=0.0034552 |
||||
z=5 P=0.0009137 |
||||
z=6 P=0.0002428 |
||||
z=7 P=0.0000647 |
||||
z=8 P=0.0000173 |
||||
z=9 P=0.0000046 |
||||
z=10 P=0.0000012 |
||||
|
||||
q=0.3 |
||||
z=0 P=1.0000000 |
||||
z=5 P=0.1773523 |
||||
z=10 P=0.0416605 |
||||
z=15 P=0.0101008 |
||||
z=20 P=0.0024804 |
||||
z=25 P=0.0006132 |
||||
z=30 P=0.0001522 |
||||
z=35 P=0.0000379 |
||||
z=40 P=0.0000095 |
||||
z=45 P=0.0000024 |
||||
z=50 P=0.0000006 |
||||
---- |
||||
|
||||
Solving for P less than 0.1%... |
||||
|
||||
[source] |
||||
---- |
||||
P < 0.001 |
||||
q=0.10 z=5 |
||||
q=0.15 z=8 |
||||
q=0.20 z=11 |
||||
q=0.25 z=15 |
||||
q=0.30 z=24 |
||||
q=0.35 z=41 |
||||
q=0.40 z=89 |
||||
q=0.45 z=340 |
||||
---- |
||||
|
||||
== 12. Conclusion |
||||
We have proposed a system for electronic transactions without relying on trust. We started with the usual framework of coins made from digital signatures, which provides strong control of ownership, but is incomplete without a way to prevent double-spending. To solve this, we proposed a peer-to-peer network using proof-of-work to record a public history of transactions that quickly becomes computationally impractical for an attacker to change if honest nodes control a majority of CPU power. The network is robust in its unstructured simplicity. Nodes work all at once with little coordination. They do not need to be identified, since messages are not routed to any particular place and only need to be delivered on a best effort basis. Nodes can leave and rejoin the network at will, accepting the proof-of-work chain as proof of what happened while they were gone. They vote with their CPU power, expressing their acceptance of valid blocks by working on extending them and rejecting invalid blocks by refusing to work on them. Any needed rules and incentives can be enforced with this consensus mechanism. |
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Reference in new issue