Driving human capital accumulation. The role of grandparents

Contents lists available at ScienceDirect The Journal of the Economics of Ageing journal homepage: www.elsevier.com/locate/jeoa Full length article Driving human capital accumulation. The role of grandparents Mauro Maria Baldi ∗, Raffaella Coppier , Elisabetta Michetti University of Macerata, Department of Economics and Law, Macerata, Italy A R T I C L E I N F O JEL classification: J13 J14 C61 O4 Keywords: Active aging Human capital accumulation Economic growth Education of grandchildren by grandparents A B S T R A C T The role of grandparents in the education process of grandchildren is becoming increasingly important, especially in economies experiencing positive human capital growth rates over time. To investigate the main forces driving such a phenomenon and the effect on human capital accumulation and growth, in the present work we propose an overlapping generations model where the time spent by grandparents on grandchildren’s education is endogenously determined to maximize the expected utility associated with the trade-off between leisure time and grandchildren’s education. By combining analytical tools and numerical simulations, our model shows that the more time grandparents spend educating their grandchildren - quantity - and the more educated they are - quality -, the more education is transmitted from older to younger generations, who will have higher levels of human capital, which will generate higher economic growth. Such a mechanism repeats over time, generating virtuous paths that work as drivers of human capital and physical capital growth. Introduction The increasing trend of longevity, or people living longer lives, has significant implications for the roles and relationships of grandparents within families. Longer and better life expectancy means grandparents may have the opportunity to develop extended relationships with their grandchildren. This extended time together allows for a deeper connection and the sharing of more life experiences. In addition, grandparents have more time to serve as mentors and guides for their grandchildren. They can share wisdom, advice, and life lessons accumulated over the years, contributing to the personal and intellectual development of their grandchildren. Longer and better life expectancy allows grandparents to serve as positive aging role models, challenging stereotypes about aging, and demonstrating the potential for continued growth, learning, and contribution to family and society. Grandparents can actively contribute to the family in various ways. They can devote part of their time to raising their grandchildren (Baldi et al., 2025) or they can contribute to their grandchildren’s education (Baldi et al., 2024). With regard to the first channel, Baldi et al. (2025) consider the elderly can divide their available time between remaining in the workforce and caring for their grandchildren. This assumption implies that grandparents play a role in child-rearing, thereby reducing the amount of resources parents must devote to raising their children. By substituting for parental effort, grandparents’ time indirectly influences adult children’s (parents’) fertility choices through its effect on household income. The results indicate that higher effective wages in highincome economies encourage greater investment in children’s education, thereby enhancing their human capital. Regarding the decision ∗Corresponding author. E-mail addresses: [email protected] (M.M. Baldi), [email protected] (R. Coppier), [email protected] (E. Michetti). of elderly to remain in the workforce or devote their time entirely to caring for grandchildren is not determined by the level of economic development, but rather by relative wage dynamics in both highand low-income countries. Regarding the role of grandparents in grandchildren’s education, Baldi et al. (2024) consider that not only parents but also grandparents can contribute to children’s education. In such a way, they investigate how active aging can be a source of economic growth through the increase in the human capital mechanism creation. In fact, as Baldi et al. (2024) stress, elderly people who choose to give up a fixed fraction of their time may contribute to their grandchildren’s education, playing an important role in human capital accumulation, fertility, and economic growth. Likewise to Baldi et al. (2024), the present work aims to analyze the role of grandparents considering that not only parents, but also grandparents, can contribute to children’s education. Including the role of grandparents allows us to study how active aging can be a source of economic growth through the increase of the human capital of the younger generation. This work builds on Baldi et al. (2024), but extends it considering that the time grandparents devote to educating their grandchildren is not exogenous but endogenous. This analysis adds greater realism to the model and allows for reflection on the determinants of time allocation choices in old age. The theoretical literature on intergenerational transmission models has largely focused on transfers from one generation to the next, from https://doi.org/10.1016/j.jeoa.2025.100598 The Journal of the Economics of Ageing 32 (2025) 100598 Available online 27 September 2025 2212-828X/© 2025 Published by Elsevier B.V. M.M. Baldi et al. parent to child. However, in recent years, a wider range of literature has developed that examines the cases where transfers can occur between multiple generations. Early economic theories of generationto-generation capital transfers shared the claim that the outcomes of grandparents and grandchildren were correlated, but argued that this correlation passed almost entirely through the parent generation (Becker and Tomes, 1986). More precisely, economic models proposed that in the transmission of human capital, parents, in addition to passing on innate endowments to their children through genes, actively invest economic and non-economic resources to promote their children’s education, see e.g., Becker and Tomes (1986) and Goldberger (1989). The main novelty proposed in the present work consists in taking into account the role of grandparental education: as grandparents have invested resources to educate their children, they may similarly find it important to contribute with different types of capital directly to the education and well-being of their grandchildren (Bol and Kalmijn, 2016). In this context, grandparents can contribute to the welfare of their grandchildren in different ways, as for instance, through direct financial transfers or support, i.e. by helping to pay for grandchildren’s education costs, encouraging and strengthening a culture within the family that places greater value on education. As long as such investments by grandparents are made during the grandchildren’s youth, they will drive the mechanism of creation of human and social capital that the grandchildren will have in adulthood. From this perspective, the resources provided by grandparents can significantly contribute to the educational results of their grandchildren. As has been argued by Mare (2011), a model that considers only two generations may not be adequate to capture all the different ways in which family background can influence what children grow up to be (see also Pfeffer, 2014 and Solon, 2014). In particular, Mare (2011) states that most theoretical models that analyze intergenerational transmission mechanisms assume a Markov process by which resources and endowments are transmitted from one generation to the next only. However, there are many channels through which the behavior of grandparents can influence the education of their grandchildren, as, for instance, economic and financial resources (inheritance), or the direct effect that frequent and personal contact between grandparents and grandchildren might have on a society where increasing longevity means that generations overlap for longer periods. As Møllegaard and Jæger (2015) stress: grandparents may also use their cultural capital to promote grandchildren’s educational success. Research shows that children spend considerable amounts of time with grandparents during childhood (Bengtson, 2001), which means that they are exposed to a larger family environment than that provided by their parents. Grandparents may inculcate cultural capital in grandchildren via this extended-family environment, for example by providing a stimulating learning environment, organizing cultural activities (for example trips to the theater or extracurricular activities), acting as role models in ways that shape grandchildren’s educational preferences (see Kohn and Slomczynski, 1990; King and Elder, 1997). The previous considerations are supported by empirical evidence showing that ‘‘extended’’ family members, particularly grandparents, play a major role in the economic, cultural, and social conditions of grandchildren (see e.g. Bengtson, 2001). More precisely, grandparents’ resources have a direct effect on grandchildren’s achievements such as cognitive development (see e.g. Modin and Fritzell, 2009 and Ferguson and Ready, 2011) thus enhancing performance achieved in schooling (see Falbo, 2014). Finally, the presence of a grandparent proximity effect has been found in many of the places where it could be sought, such as rural China (Zeng and Xie, 2014), Sweden (Lindahl et al., 2015), Great Britain (Chan and Boliver, 2013), Germany (Hertel and Groh-Samberg, 2014), and Denmark (Boserup et al., 2016). Therefore, in this paper, we analyze the effect of the role of grandparents in the education of grandchildren. More precisely, we consider that grandparents may devote part of their time to education according to an endogenous choice mechanism that takes into account an explicit trade-off, as time devoted to grandchildren’s education is time taken away from leisure time (i.e. grandparents like to impart knowledge to grandchildren, but it is costly in terms of giving up leisure time), and this result is influenced by the evolution of the experienced human capital level over time. Grandparents’ action on grandchildren’s education goes through a dual channel consisting of a ‘‘quantitative’‘ element and a ‘‘qualitative’’ element. In fact, both the quantity of time devoted to the grandchildren’s education and the quality of this time (which we approximate with the grandparents’ education, i.e. their human capital) are relevant and enter directly as endogenous variables of the overlapping generations (OLG) model we propose.1 By taking these elements into account, we obtain a framework that can describe how and to what extent an increase in the amount of time grandparents spend on their grandchildren’s education fosters the accumulation of human capital and, thus, economic growth. Our framework aims to show that the more time grandparents spend educating their grandchildren, and the more educated they are, the more education is transmitted from older to younger generations, which will have higher levels of human capital, ending in higher economic growth. Such a mechanism repeats over time, producing virtuous paths that work as drivers of human capital and physical capital growth. Our model confirms that the optimal number of children (quantity) and their optimal education level (quality) undergo a trade-off. However, differently from previous contributions in literature, the ability of children to absorb the human capital of their parents is not dependent only on a spillover effect, but also on the contribution given by grandparents to the education of grandchildren. Furthermore, when considering the ratio between the human capital and physical capital growth rates, we show that it results to be positively correlated to the time grandparents spend educating their grandchildren. Finally, conditions for the existence of a balanced growth path in which the human capital and physical capital grow at the same rate, are obtained. However, numerical experiments show that it is not stable since, depending on initial conditions and on parameter values, the economy may converge in the long run to two different final scenarios, namely the best scenario (the ratio between human capital and physical capital growth rates increases over time and grandparents devote all their time to educating grandchildren) and the worst scenario (the ratio between human capital and physical capital growth rates converges and grandparents do not devote all their time to educating grandchildren). Since economies starting from a high level of initial development may fall into a poverty trap, i.e. ending in the worst scenario, a subsistence trap may emerge. However, by means of numerical experiments, we show that, if the human capital initial level is high enough, or the importance grandparents give to leisure time is low enough or, finally, if the productivity of parental educational expenditure in the production of human capital is high enough, then the subsistence trap may be avoided. The paper proceeds as follows. Section ‘‘Model setup’’ illustrates the setup of the model. Section ‘‘Discussion’’ presents both the analytical and numerical results. Section ‘‘Conclusion’’ concludes the paper. Model setup We consider a standard OLG model populated by agents living for three periods. Time is discrete and denoted by 𝑡∈N, while the length of each period is normalized to one. Children receive education in the first period, adults work and raise children in the second, and old people retire in the third and spend part of their time on grandchildren’s education. 1For a thorough analysis of overlapping generations models that can serve as a workhorse for policy analysis, see e.g. Gorokhovsky and Rubinchik (2024). The Journal of the Economics of Ageing 32 (2025) 100598 2 M.M. Baldi et al. At time 𝑡 the economy consists of 𝑁𝑡−1 old agents, 𝑁𝑡 adult agents and 𝑁𝑡+1 young agents. Assuming that adults at time 𝑡 raise 𝑛𝑡 children, the number of adult agents at time 𝑡+ 1 evolves according to the following rule: 𝑁𝑡+1 =𝑛𝑡𝑁𝑡,(1) where 𝑛𝑡 represents the number of children per adult at time 𝑡. Adults choose how many resources to devote to educating and raising children. Parents are ‘‘altruistic’’, as they care, not only, about consumption and leisure, but also about their children and the human capital of their children (Becker and Tomes, 1976). The innovative element of this work is the role played by grandparents in the education of grandchildren. Indeed, in their retirement period, the elderly can allocate their available unit of time between leisure time and the education of grandchildren. The active role of grandparents is emphasized in our model: each child will be raised by parents and will be educated by both parents and grandparents. The time allocated by grandparents to educating grandchildren is endogenous and determined within the model by considering the explicit trade-offs between time devoted to educating grandchildren and leisure time. Hence, the choice made by adults on how much time to spend both on leisure and also to educate grandchildren when old is explicitly added to the utility maximization problem. Lifetime utility maximization To take into account the decision about how much time to spend on the education of grandchildren when old, we consider that, at time 𝑡, the adult maximizes the utility function by incorporating new nonnegative decision variables. These are: the leisure time fraction when old, 𝑙𝑡+1, and the time fraction devoted to educating grandchildren during retirement, 𝜇𝑡+1, with 𝑙𝑡+1 +𝜇𝑡+1 = 1. The importance attached to education time is strictly related to the evolution of human capital, in the sense that the importance associated with the educational component becomes more relevant as long as increasing levels of human capital are observed, which are measured by the ratio ℎ𝑡 ℎ𝑡−1 providing that the attitude to spending time on education increases when higher levels of human capital growth are reached. Recall that ℎ𝑡 is the level of education of the adult at time 𝑡 and ℎ𝑡−1 is the level of education of the parent of the adult. The relationship between these two quantities provides us with a proxy for the growth of the adult subject’s education relative to his or her parents. So, if the adult has received more education than his parents, he will give greater importance to education (a sort of inherited attitude, see Gori and Michetti, 2016 and Fanti et al., 2019) and thus greater importance to the time to be devoted, when old, to educating his grandchildren. On the other hand, we denote with 𝜃 > 0 the weight associated to leisure time, that in this framework is assumed to be constant. In summary, adults derive utility from the consumption they will enjoy in the adulthood period and in old age, the number of children and their ‘‘quality’’ (human capital), from the social interactions enjoyed during the time left educating children in old age, which hereafter we will call ‘‘leisure time’’, and, finally, from the time devoted to educating their grandchildren. Starting from Baldi et al. (2024), the utility function of an adult becomes: 𝑈𝑡= ln 𝑐1,𝑡 +𝜌ln 𝑐2,𝑡+1 +𝜃ln (𝑙𝑡+1) + ℎ𝑡 ℎ𝑡−1 ln 𝜇𝑡+1 +𝛽ln ℎ𝑡+1 +𝜆ln 𝑛𝑡,(2) where 𝑐1,𝑡 represents consumption during adulthood age, 𝑐2,𝑡+1 is consumption during retirement, 𝑛𝑡 is the number of children, ℎ𝑡+1 is the human capital of the offspring, 𝜇𝑡+1 and 𝑙𝑡+1 are the fraction of available time that adults plan to devote to educating grandchildren and to leisure when they are elderly, respectively, and all variables are not negative. The parameter 𝜌∈ (0,1) represents the discount factor, 𝜆∈ (0,1) and 𝛽∈ (0,1) represent the importance attached to quantity and quality of the children, respectively, 𝜃 > 0 is the weight given to leisure time while ℎ𝑡 ℎ𝑡−1 is the human capital growth rate from time 𝑡− 1 to time 𝑡 representing the weight given to the time devoted to educating grandchildren.2 We denote with 𝑤𝑡 the wage rate for a unit of effective labor, 𝑠𝑡 the life-cycle saving, and 𝑒𝑡 the per-child educational expenditure. Furthermore, following Hirazawa and Yakita (2017), 𝑧∈ (0,1) is the child-rearing cost per child, which is assumed to be equal to a constant proportion of the wage income. Then, given the parents’ income, the cost of raising children and the cost of education distract resources from alternative uses, i.e., present and future consumption. Therefore, budget constraints for the adult agent result as follows. 𝑤𝑡ℎ𝑡=𝑐1,𝑡 +𝑠𝑡+𝑛𝑡(𝑒𝑡+𝑧𝑤𝑡ℎ𝑡).(3) As usual, consumption of the old period is financed by returns to the savings accumulated during adulthood (Galor and Weil, 2000).3 Old agents rent their saving of time 𝑡 as capital to firms at time 𝑡+ 1, and receive the interest factor 𝑟𝑡+1. Therefore, the budget constraint in old age will be 𝑐2,𝑡+1 =𝑟𝑡+1𝑠𝑡.(4) Following Baldi et al. (2024), we assume that the education of children consists of three parts: the investment in education by parents, the time devoted by grandparents to educating grandchildren, and the average level of human capital in the economy. The ‘‘parental’’ component is based on the fact that the education of offspring is entrusted to educational institutions, i.e., parents delegate, by paying, the formal education of their offspring to the educational system (De La Croix and Doepke, 2004). The component that comes from grandparents represents both a ‘‘quantity’’ factor, i.e., the time grandparents devote to each grandchild (𝜇𝑡 𝑛𝑡 ), and a ‘‘quality’’ factor of time which we approximate by grandparent education, i.e. their human capital ℎ𝑡−1. Differently from Baldi et al. (2024), we consider that there is less than perfect substitutability between grandparents and the school in the production of human capital. Finally, the average level of human capital in the economy,  ℎ𝑡, also plays a role due to spillover effects, i.e., due to the fact that the learning process is more effective if a child interacts with more educated individuals. Therefore, the human capital production function will be ℎ𝑡+1 =(𝜋+𝜙𝑒𝑡)𝜏(ℎ𝑡−1 𝜇𝑡 𝑛𝑡)𝛿  ℎ1−𝛿 𝑡, 𝛿 ∈ (0,1) 𝜏∈ (0,1),(5) where ℎ𝑡−1 is the stock of human capital of the old generation, 𝑒𝑡 is the educational expenditure. The parameters satisfy 𝜋, 𝜙 > 0. As usual in literature, the presence of 𝜋∈ (0,1] guarantees that human capital remains positive also in the case in which parents do not invest in education. Finally, 𝜏∈ (0,1) indicates the importance of the role of ‘‘parents’’ in the education of the children, (1 − 𝛿) ∈ (0,1) is the productivity of the average level of human capital in the economy, so that 𝛿 expresses the importance of grandparents’ education transmission mechanism. Finally, the following unit time constraints must be added. In fact, supposing one unit of disposable time to be shared between leisure and grandchildren’s education when old, it must be 𝜇𝑡+1 +𝑙𝑡+1 = 1.(6) 2In our model we do not consider the fact that individuals take care also the minimum utility across time. For an analysis of the role of wariness in an OLG model, see e.g. Pham and Pham (2024). 3In our model, we simplify the analysis by considering that old agents can live only with the amount of resources saved when young, avoiding pension benefits (for a detailed analysis of the dynamic properties of an OLG economy with endogenous fertility and fertility-related pensions see e.g. Hirazawa and Yakita, 2009; Fanti and Gori, 2012, 2013). The Journal of the Economics of Ageing 32 (2025) 100598 3 M.M. Baldi et al. The problem for the adult generation at time 𝑡 is to choose how much to consume 𝑐1,𝑡 and, hence, the saving level 𝑠𝑡 or the consumption level when old 𝑐2,𝑡+1, the number of children 𝑛𝑡, the parental educational expenditure 𝑒𝑡 and how to share disposable time between educating grandchildren and leisure when old, i.e. 𝜇𝑡+1 and 𝑙𝑡+1, in order to maximize the lifetime utility 𝑈𝑡 under budget constraints. Notice that ℎ𝑡= ℎ𝑡 from the assumption of identical individuals within a generation. The related maximization problem is as follows: max 𝑈𝑡(𝑐1,𝑡, 𝑐2,𝑡+1, 𝜇𝑡+1, 𝑙𝑡+1, 𝑒𝑡, 𝑛𝑡) = = ln 𝑐1,𝑡 +𝜌ln 𝑐2,𝑡+1 +𝜃ln (𝑙𝑡+1) + ℎ𝑡 ℎ𝑡−1 ln 𝜇𝑡+1 +𝛽ln ℎ𝑡+1 +𝜆ln 𝑛𝑡 (7) s. t.: 𝑤𝑡ℎ𝑡=𝑐1,𝑡 +𝑠𝑡+𝑛𝑡(𝑒𝑡+𝑧𝑤𝑡ℎ𝑡)(8) 𝑐2,𝑡+1 =𝑟𝑡+1𝑠𝑡(9) ℎ𝑡+1 =(𝜋+𝜙𝑒𝑡)𝜏(ℎ𝑡−1 𝜇𝑡 𝑛𝑡)𝛿  ℎ1−𝛿 𝑡(10) 𝑙𝑡+1 +𝜇𝑡+1 = 1.(11) Necessary and sufficient conditions applied to the constrained maximization problem give the following solutions (see Appendix A for a sketch of the proof).4 Proposition 1. Let 𝑀=𝜆−𝛽𝛿 𝛽𝜏 and 𝜆≠𝛽(𝛿+𝜏). The first order (necessary) conditions for model (7)–(11) are as follows: 𝑐1, 𝑡 =𝑤𝑡ℎ𝑡 1 + 𝜌+𝜆−𝛽𝛿 ,(12) 𝑐2, 𝑡+1 =𝜌𝑟𝑡+1𝑤𝑡ℎ𝑡 1 + 𝜌+𝜆−𝛽𝛿 ,(13) 𝜇𝑡+1 =ℎ𝑡 ℎ𝑡+𝜃ℎ𝑡−1 ,(14) 𝑙𝑡+1 =𝜃ℎ𝑡−1 ℎ𝑡+𝜃ℎ𝑡−1 ,(15) 𝑒𝑡=⎧ ⎪ ⎨ ⎪ ⎩ 𝛽𝜏 𝜆−𝛽(𝛿+𝜏)𝑧𝑤𝑡ℎ𝑡−𝜆−𝛽𝛿 𝜆−𝛽(𝛿+𝜏) 𝜋 𝜙if 𝛽𝜏 𝜆−𝛽(𝛿+𝜏)𝑧𝑤𝑡ℎ𝑡−𝜆−𝛽𝛿 𝜆−𝛽(𝛿+𝜏) 𝜋 𝜙≥0 0if 𝑤𝑡ℎ𝑡≤𝜋 𝜙𝑧 𝑀, (16) 𝑛𝑡=⎧ ⎪ ⎨ ⎪ ⎩ 𝜆−𝛽(𝛿+𝜏) 1+𝜌+𝜆−𝛽𝛿 𝜙𝑤𝑡ℎ𝑡 𝑧𝜙𝑤𝑡ℎ𝑡−𝜋if 𝛽𝜏 𝜆−𝛽(𝛿+𝜏)𝑧𝑤𝑡ℎ𝑡−𝜆−𝛽𝛿 𝜆−𝛽(𝛿+𝜏) 𝜋 𝜙≥0 𝜆−𝛽𝛿 1+𝜌+𝜆−𝛽𝛿 1 𝑧if 𝑤𝑡ℎ𝑡≤𝜋 𝜙𝑧 𝑀. (17) Moreover, the second order (sufficient) conditions for model (7)–(11) are as follows: 𝜆 > 𝛽(𝛿+𝜏)(18) and 𝑤𝑡ℎ𝑡>𝜋 𝜙𝑧 √𝑀. (19) By combining the first and second order conditions, we can rewrite Eqs. (12)–(17) as follows: 𝑐1, 𝑡 =𝑤𝑡ℎ𝑡 1 + 𝜌+𝜆−𝛽𝛿 ,(20) 𝑐2, 𝑡+1 =𝜌𝑟𝑡+1𝑤𝑡ℎ𝑡 1 + 𝜌+𝜆−𝛽𝛿 ,(21) 4Verifying not only first-order but also second-order conditions is not an easy task due to the analytical complexity of the problem. Many authors avoid the verification of sufficient conditions. We emphasize the robustness of our solutions, which makes the mathematical proof an important part of the proposed study. 𝜇𝑡+1 =ℎ𝑡 ℎ𝑡+𝜃ℎ𝑡−1 ,(22) 𝑙𝑡+1 =𝜃ℎ𝑡−1 ℎ𝑡+𝜃ℎ𝑡−1 ,(23) 𝑒𝑡=⎧ ⎪ ⎨ ⎪ ⎩ 𝛽𝜏 𝜆−𝛽(𝛿+𝜏)𝑧𝑤𝑡ℎ𝑡−𝜆−𝛽𝛿 𝜆−𝛽(𝛿+𝜏) 𝜋 𝜙if 𝑤𝑡ℎ𝑡>𝜋 𝜙𝑧 𝑀 0if 𝜋 𝜙𝑧 √𝑀 < 𝑤𝑡ℎ𝑡≤𝜋 𝜙𝑧 𝑀, (24) 𝑛𝑡=⎧ ⎪ ⎨ ⎪ ⎩ 𝜆−𝛽(𝛿+𝜏) 1+𝜌+𝜆−𝛽𝛿 𝜙𝑤𝑡ℎ𝑡 𝑧𝜙𝑤𝑡ℎ𝑡−𝜋if 𝑤𝑡ℎ𝑡>𝜋 𝜙𝑧 𝑀 𝜆−𝛽𝛿 1+𝜌+𝜆−𝛽𝛿 1 𝑧if 𝜋 𝜙𝑧 √𝑀 < 𝑤𝑡ℎ𝑡≤𝜋 𝜙𝑧 𝑀. (25) Notice that, from a purely mathematical point of view, the second-order conditions (such as, for example, 𝜆 > 𝛽(𝛿+𝜏)) arise from satisfying the requirement that the Hessian matrix must be negative definite on the set, defined by the equality and active inequality constraints. This condition may also have an economic interpretation. Recalling that 𝛽 represents the weight parents assign to their children’s human capital in the utility function, 𝛿 denotes the productivity of education 𝑒𝑡 of the parents in the production of human capital, and 𝜏 represents the productivity of education of the grandparents in the production of human capital, then the term 𝛽(𝛿+𝜏) captures the utility parents derive from the human capital generated through their investment in education and those of their grandparents. In addition, 𝜆 is the weight attached to the number of children, so condition 𝜆 > 𝛽(𝛿+𝜏) states that the parental utility associated with having children is greater than the utility associated with human capital. Notice that this condition is very often used in literature (see e.g., Yakita, 2010). Furthermore, from Proposition 1, it can be easily observed that the set of admissible solutions and satisfying second-order conditions is divided into two regions, which we will conveniently call region A and region B. In particular, from this point onward, we assume that 𝜆 > 𝛽(𝛿+𝜏) and denote by region A the set of admissible solutions such that 𝑤𝑡ℎ𝑡>𝜋 𝜙𝑧 𝑀. Likewise, we denote by region B the set of admissible solutions such that 𝜋 𝜙𝑧 √𝑀 < 𝑤𝑡ℎ𝑡≤𝜋 𝜙𝑧 𝑀. We can note that 𝑑𝑛𝑡 𝑑𝑤𝑡 <0 and, 𝑑𝑒𝑡 𝑑𝑤𝑡 >0 for 𝑤𝑡ℎ𝑡>𝜋𝑀 𝜙𝑧 , therefore, this labor income is a threshold effective wage rate at which parents find it worthwhile to invest in the education of children, as it occurs in region 𝐴. The economic intuition underlying this threshold value can be derived by comparing the marginal benefit/marginal cost ratio of raising a child with the marginal benefit/marginal cost ratio of educating a child. More precisely, the marginal cost of an additional child is given by 𝑤𝑡ℎ𝑡𝑧, while the marginal benefit is given by 𝑑𝑈𝑡 𝑑𝑛𝑡 = 𝜆−𝛽𝛿 𝑛𝑡 at 𝑒𝑡= 0 and, consequently, the ratio marginal benefit on marginal cost is: 𝜆−𝛽𝛿 𝑛𝑡𝑤𝑡ℎ𝑡𝑧. Regarding the marginal benefit/marginal cost ratio of educating children, the marginal benefit is given by 𝑑𝑈𝑡 𝑑ℎ𝑡+1 𝑑ℎ𝑡+1 𝑑𝑒𝑡 =𝛽𝜃𝜙 𝜋 at 𝑒𝑡= 0. The marginal cost of educating children is 𝑑(𝑒𝑡𝑛𝑡) 𝑑𝑒𝑡 =𝑛𝑡. Therefore, the benefit/marginal cost ratio of educating children is equal to 𝛽𝜃𝜙 𝜋𝑛𝑡 . If the former ratio is greater than the latter, then parents prefer to have another child rather than invest in the education of their child. In the opposite case, when the wage income is less than this labor income threshold, the parents prefer to invest their wage in the education of children rather than have more children. As in De La Croix and Doepke (2003), the classical trade-off between the education of children (quality) and the number of children (quantity) emerges. Both fertility 𝑛𝑡 and education 𝑒𝑡 as a function of wage income 𝑤𝑡ℎ𝑡 are represented in Fig. 1. As in De La Croix and Doepke (2003, 2004), fertility depends negatively on the effective wage, while education is increasing with respect to the effective wage. Production As usual, we consider an economic framework made up of many competitive firms operating with constant returns to scale in their The Journal of the Economics of Ageing 32 (2025) 100598 4 M.M. Baldi et al. Fig. 1. 𝑒𝑡 and 𝑛𝑡 as a function of 𝑤𝑡ℎ𝑡. production technology. It is also assumed that the capital stock fully depreciates after one period of use, meaning that the capital stock in any given period equals the savings from the previous period. Production at time 𝑡 uses physical capital 𝐾𝑡 and labor 𝐿𝑡. We denote the aggregate output as 𝑄𝑡 and describe the aggregate technology with the following Cobb–Douglas production function: 𝑄𝑡=𝐹(𝐾𝑡, 𝐿𝑡) = 𝐴𝐾𝛼 𝑡𝐿1−𝛼 𝑡,(26) where 𝐴 > 0 represents the total factor productivity and 𝛼∈(0,1) indicates the productivity of physical capital. The firm selects its input levels by maximizing profits 𝑄𝑡−𝑤𝑡𝐿𝑡−𝑟𝑡𝐾𝑡. The profit maximization implies that the wage rate and the interest factor are equal to the marginal productivity of labor and capital, so that: 𝑤𝑡∶= 𝐴(1 − 𝛼)𝐾𝛼 𝑡 𝐿𝑡𝛼(27) and 𝑟𝑡∶= 𝐴𝛼𝐾𝛼−1 𝑡 𝐿𝑡𝛼−1 .(28) Following Hirazawa and Yakita (2017), the effective labor 𝐿𝑡 is given by 𝐿𝑡=ℎ𝑡𝑁𝑡. Define 𝑘𝑡=𝐾𝑡 𝑁𝑡 . In this light, formulae (27) and (28) become 𝑤𝑡=𝐴(1 − 𝛼)𝑘𝛼 𝑡 ℎ𝛼 𝑡 (29) and 𝑟𝑡=𝐴𝛼 𝑘𝛼−1 𝑡 ℎ𝛼−1 𝑡 .(30) Moreover, since we assume 𝐾𝑡+1 =𝑠𝑡𝑁𝑡, 𝑁𝑡+1 =𝑛𝑡𝑁𝑡, we have that 𝑘𝑡+1 =𝐾𝑡+1 𝑁𝑡+1 =𝑠𝑡𝑁𝑡 𝑛𝑡𝑁𝑡 =𝑠𝑡 𝑛𝑡 .(31) Market equilibrium The dynamical systems associated with regions A and B and describing the evolution of capital per capita and human capital are stated in the following proposition, whose proof is in Appendix B. Proposition 2. Let 𝜆 > 𝛽(𝛿+𝜏) and define 𝑣𝑡=ℎ𝑡−1, then the dynamical systems associated to the two regions coming from Proposition 1 can be formulated as follows. •Case A: if 𝑘𝛼 𝑡ℎ1−𝛼 𝑡>𝜋𝑀 𝜙𝑧𝐴(1−𝛼): 𝑇𝐴∶= ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 𝑘𝑡+1 =𝜌 𝜆−𝛽𝛿−𝛽𝜏 [𝑧𝐴(1 − 𝛼)𝑘𝛼 𝑡ℎ1−𝛼 𝑡−𝜋 𝜙] ℎ𝑡+1 =(𝛽𝜏𝜙)𝜏(1+𝜌+𝜆−𝛽𝛿)𝛿 (𝜆−𝛽𝛿−𝛽𝜏)𝛿+𝜏𝐴𝛿(1−𝛼)𝛿𝑘−𝛼𝛿 𝑡ℎ1−2𝛿+𝛼𝛿 𝑡𝜇𝛿 𝑡𝑣𝛿 𝑡[𝑧𝐴(1 − 𝛼)𝑘𝛼 𝑡ℎ1−𝛼 𝑡−𝜋 𝜙]𝛿+𝜏 𝜇𝑡+1 =1 1+𝜃𝑣𝑡 ℎ𝑡 𝑣𝑡+1 =ℎ𝑡 . (32) •Case B: if 𝜋√𝑀 𝜙𝑧𝐴(1−𝛼)< 𝑘𝛼 𝑡ℎ1−𝛼 𝑡≤𝜋𝑀 𝜙𝑧𝐴(1−𝛼): 𝑇𝐵∶= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 𝑘𝑡+1 =𝜌 𝜆−𝛽𝛿 𝑧𝐴(1 − 𝛼)𝑘𝛼 𝑡ℎ1−𝛼 𝑡 ℎ𝑡+1 =𝜋𝜏[(1+𝜌+𝜆−𝛽𝛿)𝑧 𝜆−𝛽𝛿 ]𝛿 𝜇𝛿 𝑡ℎ1−𝛿 𝑡𝑣𝛿 𝑡 𝜇𝑡+1 =1 1+𝜃𝑣𝑡 ℎ𝑡 𝑣𝑡+1 =ℎ𝑡 .(33) Systems 𝑇𝐴 and 𝑇𝐵 describe the evolution of physical capital per capita 𝑘𝑡 and human capital ℎ𝑡 over time, and, in addition, also the dynamics of the education time are endogenously described by the model. We observe that the dynamics of the state variables are not described as long as 𝑘𝛼 𝑡ℎ1−𝛼 𝑡≤𝜋√𝑀 𝜙𝑧𝐴(1−𝛼). From a mathematical point of view, this emerges from the second-order conditions related to the constrained optimization problem (see Appendix A) while, from an economic point of view, it can be interpreted by the fact that a minimum level or subsistence level of wage income is needed to guarantee a decorous living standard. Hence, we assume that there exists a minimum level of effective living wage, namely 𝑤𝑚ℎ𝑚=𝜋√𝑀 𝜙𝑧 , representing a lower bound of the effective wage rate 𝑤𝑡ℎ𝑡. As a consequence, ∀ 0 ≤𝑤𝑡ℎ𝑡≤𝑤𝑚ℎ𝑚 we assume 𝑘𝛼 𝑡ℎ1−𝛼 𝑡=𝜋√𝑀 𝜙𝑧𝐴(1 − 𝛼), thus implying that, in the open case 𝑘𝛼 𝑡ℎ1−𝛼 𝑡≤𝜋√𝑀 𝜙𝑧𝐴(1−𝛼), the dynamics of state variables can be described by the following system 𝑇𝐶: 𝑇𝐶∶= ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 𝑘𝑡+1 =𝜌 𝜆−𝛽𝛿 𝜋√𝑀 𝜙 ℎ𝑡+1 =𝜋𝜏[(1+𝜌+𝜆−𝛽𝛿)𝑧 𝜆−𝛽𝛿 ]𝛿 𝜇𝛿 𝑡ℎ1−𝛿 𝑡𝑣𝛿 𝑡 𝜇𝑡+1 =1 1+𝜃𝑣𝑡 ℎ𝑡 𝑣𝑡+1 =ℎ𝑡 .(34) The system 𝑇𝐶 ensures the existence of a minimum level of capital per capita that can be produced in the economy. The Journal of the Economics of Ageing 32 (2025) 100598 5 M.M. Baldi et al. To illustrate how grandparents’ involvement in their grandchildren’s education influences the long-term dynamics of the system and the child Q-Q trade-off, we review the equilibrium conditions for the number of children 𝑛𝑡 and education 𝑒𝑡, derived from the constraints of the maximization problem. Thus, the optimal number of children at any time 𝑡 and the optimal education level can be derived by substituting (29) into (24) and (25). As a result, the quality 𝑒𝑡 and quantity 𝑛𝑡 are obtained as follows: 𝑒𝑡=⎧ ⎪ ⎨ ⎪ ⎩ 𝛽𝜏 𝜆−𝛽(𝛿+𝜏)𝑧𝐴(1 − 𝛼)𝑘𝛼 𝑡ℎ1−𝛼 𝑡−𝜆−𝛽𝛿 𝜆−𝛽(𝛿+𝜏) 𝜋 𝜙if 𝐴(1 − 𝛼)𝑘𝛼 𝑡ℎ1−𝛼 𝑡>𝜋 𝜙𝑧 𝑀 0if 0≤𝐴(1 − 𝛼)𝑘𝛼 𝑡ℎ1−𝛼 𝑡≤𝜋 𝜙𝑧 𝑀, (35) 𝑛𝑡=⎧ ⎪ ⎨ ⎪ ⎩ 𝜆−𝛽(𝛿+𝜏) 1+𝜌+𝜆−𝛽𝛿 𝜙𝐴(1−𝛼)𝑘𝛼 𝑡ℎ1−𝛼 𝑡 𝑧𝜙𝐴(1−𝛼)𝑘𝛼 𝑡ℎ1−𝛼 𝑡−𝜋if 𝐴(1 − 𝛼)𝑘𝛼 𝑡ℎ1−𝛼 𝑡>𝜋 𝜙𝑧 𝑀 𝜆−𝛽𝛿 1+𝜌+𝜆−𝛽𝛿 1 𝑧if 0≤𝐴(1 − 𝛼)𝑘𝛼 𝑡ℎ1−𝛼 𝑡≤𝜋 𝜙𝑧 𝑀. (36) In following Proposition 3, we establish a property regarding the value of 𝑛𝑡 (the proof is in Appendix C). Proposition 3. The maximum value of 𝑛𝑡 is 𝜆−𝛽𝛿 1+𝜌+𝜆−𝛽𝛿 1 𝑧. Our model confirms the classic Q-Q trade-off; for high wage income, parents find it worthwhile to educate their children and raise fewer of them; conversely, when the wage income is low, parents do not educate their children but find it optimal to generate them. However, in both cases, unlike Yakita (2010), the education of children does not depend on a spillover effect (the ability of children to absorb the human capital of their parents) but on the time spent by grandparents on the education of grandchildren. More precisely, when wage incomes are high, as the amount of time grandparents spend with grandchildren increases, the parental education – that is partially substituted – is reduced, but this frees up more resources that lead parents, ceteris paribus, to raise more children. Discussion Grandchildren education by grandparents and growth From systems 𝑇𝐴 and 𝑇𝐵 some considerations can be stated. In particular, there exists a virtuous circle linking human capital growth with physical capital growth through the intervention of grandparents in educating grandchildren. In fact, even if the education time by the elderly does not enter into the physical capital evolution law, as long as 𝜇𝑡 increases, then ℎ𝑡+1 increases and, hence, the capital per capita level in the next time increases too. Such evidence can be easily observed by computing 𝜕𝑘𝑡+1 𝜕ℎ𝑡 and 𝜕ℎ𝑡+1 𝜕𝜇𝑡 and seeing that they are always positive in both regions A and B. In fact, in region A we have: 𝜕𝑘𝑡+1 𝜕ℎ𝑡 =𝜌 𝜆−𝛽𝛿 −𝛽𝜃 𝑧𝐴(1 − 𝛼)2𝑘𝛼 𝑡ℎ−𝛼 𝑡>0 and 𝜕ℎ𝑡+1 𝜕𝜇𝑡 =(𝛽𝜏𝜙)𝜏(1 + 𝜌+𝜆−𝛽𝛿)𝛿 (𝜆−𝛽𝛿 −𝛽𝜏)𝛿+𝜏𝐴𝛿(1 − 𝛼)𝛿𝑘−𝛼𝛿 𝑡ℎ1−2𝛿+𝛼𝛿 𝑡𝛿𝜇𝛿−1 𝑡𝑣𝛿 𝑡[𝑧𝐴(1 − 𝛼)𝑘𝛼 𝑡ℎ1−𝛼 𝑡−𝜋 𝜙]𝛿+𝜏 >0. Similarly, in region B we have: 𝜕𝑘𝑡+1 𝜕ℎ𝑡 =𝜌 𝜆−𝛽𝛿 𝑧𝐴(1 − 𝛼)2𝑘𝛼 𝑡ℎ−𝛼 𝑡>0 and 𝜕ℎ𝑡+1 𝜕𝜇𝑡 =𝜋𝜏[(1 + 𝜌+𝜆−𝛽𝛿)𝑧 𝜆−𝛽𝛿 ]𝛿 𝛿𝜇𝛿−1 𝑡ℎ1−𝛿 𝑡𝑣𝛿 𝑡>0. The balanced growth path In this section, we will study the balanced growth path, i.e., a situation in which the main economic variables, such as physical and human capital, and output grow at constant rates over the long term, maintaining stable proportions between them. In relation to this path, we will also try to analyze whether this balanced growth path is stable i.e., whether the dynamic system generates trajectories converging to this path in the long run. Before proceeding with the analysis, we wish to emphasize that we have numerically observed that both variables (human capital and physical capital) exhibit increasing trajectories over time. These results are unsurprising in OLG models. It therefore becomes crucial to focus on the study of new variables that explain the relationship between the growth rates of each original variable. Thus, to describe the dynamic behavior of the economy, it is useful to introduce the human capital and physical capital growth rates. The expression for the growth rate of average human capital is 𝑔ℎ,𝑡+1 =ℎ𝑡+1 −ℎ𝑡 ℎ𝑡 ,∀𝑡∈N, while for the capital per capita we have 𝑔𝑘,𝑡+1 =𝑘𝑡+1 −𝑘𝑡 𝑘𝑡 ,∀𝑡∈N. We recall that along the balanced growth path, the growth rate of average human capital equals the growth rate of capital per capita, i.e. 𝑔ℎ,𝑡+1 =𝑔𝑘,𝑡+1 ⟹ ℎ𝑡+1 ℎ𝑡 =𝑘𝑡+1 𝑘𝑡 ,∀𝑡∈N. Let 𝑔𝑡+1 represents the ratio of the human capital growth rate to the physical capital growth rate, i.e.: 𝑔𝑡+1 =ℎ𝑡+1 ℎ𝑡 𝑘𝑡 𝑘𝑡+1 ,(37) then along the balanced growth the following relationship holds: 𝑔𝑡+1 = 1. Following Kitaura and Yakita (2010), in the following we assume that 𝛿= (1 − 𝜏) since our main concern is balanced growth. The following Proposition holds. The proof is in Appendix D. Proposition 4. Let 𝜏= 1 − 𝛿 and consider systems 𝑇𝐴 and 𝑇𝐵 as defined in Proposition 2. The following expressions then hold: •Case A: 𝑔𝐴,𝑡+1 =[𝛽(1 − 𝛿)𝜙]1−𝛿 𝜌[1 + 𝜌+𝜆−𝛽𝛿 𝐴(1 − 𝛼)]𝛿 𝑘1−𝛼𝛿 𝑡ℎ−𝛿(2−𝛼) 𝑡𝜇𝛿 𝑡𝑣𝛿 𝑡. •Case B: 𝑔𝐵,𝑡+1 =[𝜋(𝜆−𝛽𝛿)]1−𝛿[(1 + 𝜌+𝜆−𝛽𝛿)𝑧]𝛿 𝜌𝑧𝐴(1 − 𝛼)𝑘1−𝛼 𝑡ℎ−1+𝛼−𝛿 𝑡𝜇𝛿 𝑡𝑣𝛿 𝑡. We can refer to this variable as the ratio between the human capital and physical capital growth rates. In both cases, these ratios of human and physical capital growth rates increase as the time grandparents spend educating their grandchildren increases. It is also interesting to note the role of the cost of raising children, i.e. 𝑧. This cost is present in the growth rate characterizing an economy with an intermediate wage income level (Case B) because it is the situation in which parents decide not to educate their children but to increase their number: therefore, in such a scenario, the human capital of the children is lower because the contribution of the parents is missing, since raising many children takes away high resources from savings. By taking into account Proposition 4 and Eq. (37), then the following Proposition trivially holds. The Journal of the Economics of Ageing 32 (2025) 100598 6 M.M. Baldi et al. Proposition 5. Let 𝜏= 1 − 𝛿 and define 𝐶𝐴=[𝛽(1 − 𝛿)𝜙]1−𝛿 𝜌[1 + 𝜌+𝜆−𝛽𝛿 𝐴(1 − 𝛼)]𝛿 and 𝐶𝐵=[𝜋(𝜆−𝛽𝛿)]1−𝛿[(1 + 𝜌+𝜆−𝛽𝛿)𝑧]𝛿 𝜌𝑧𝐴(1 − 𝛼). Then, along the balanced growth path 𝑔𝑡+1 = 1, the following relations hold. •Case A: if 𝑘𝛼 𝑡ℎ1−𝛼 𝑡>𝜋𝑀 𝜙𝑧𝐴(1−𝛼), then 𝑘𝑡=[1 𝐶𝐴 ℎ𝛿(1−𝛼) 𝑡(ℎ𝑡 𝜇𝑡𝑣𝑡)𝛿]1 1−𝛼𝛿 .(38) •Case B: if 𝜋√𝑀 𝜙𝑧𝐴(1−𝛼)< 𝑘𝛼 𝑡ℎ1−𝛼 𝑡≤𝜋𝑀 𝜙𝑧𝐴(1−𝛼), then 𝑘𝑡=ℎ𝑡[1 𝐶𝐵(ℎ𝑡 𝜇𝑡𝑣𝑡)𝛿]1 1−𝛼 .(39) Notice that, since 𝑣𝑡 ℎ𝑡 =1 − 𝜇𝑡+1 𝜃𝜇𝑡+1 , then (ℎ𝑡 𝜇𝑡𝑣𝑡)=𝜇𝑡+1 𝜇𝑡 𝜃 1 − 𝜇𝑡+1 . Therefore, in both cases, A and B, physical capital, as already mentioned, grows as human capital grows, but it also grows as the amount of time grandparents spend educating their grandchildren increases. We also observe that, if 𝑘𝛼 𝑡ℎ1−𝛼 𝑡≤𝜋√𝑀 𝜙𝑧𝐴(1−𝛼) then system (34) must be taken into account and consequently 𝑘𝑡 𝑘𝑡+1 = 1 thus implying that 𝑔𝐶,𝑡+1 =ℎ𝑡+1 ℎ𝑡 =𝜋(1−𝛿)[(1 + 𝜌+𝜆−𝛽𝛿)𝑧 𝜆−𝛽𝛿 ]𝛿 𝜇𝛿 𝑡ℎ−𝛿 𝑡𝑣𝛿 𝑡,(40) hence, along the balanced growth path, ℎ𝑡=𝜋 (1−𝛿) 𝛿[(1 + 𝜌+𝜆−𝛽𝛿)𝑧 𝜆−𝛽𝛿 ]𝜇𝑡𝑣𝑡. A more complex question is related to the stability of the balance growth path, i.e. if the dynamical system produces trajectories converging to it in the long run. Such a question can be addressed only by using numerical experiments. Numerical experiments Being 𝜇𝑡 an endogenous variable, the resulting model is very difficult to be investigated from an analytical point of view due to its complexity (non-linearity) and the high number of variables and parameters involved. Hence we have to proceed in terms of numerical experiments to show the main features of the system. We underline that even empirical evidence does not represent a result, since we made a high number of numerical experiments, we report in what follows the main findings arising from our experiments. In Table 1, we recall the parameters involved in the model together with their range and, when available, the most common values used in literature. In the following calibration, we set the parameters constellation as follows: 𝜆= 0.3, 𝛽= 0.169, 𝛿= 0.6, 𝜌= 0.2994, 𝑧= 0.075, 𝛼= 0.25, 𝐴= 1.01, 𝜋= 0.2, 𝜙= 1. In our framework, 𝜏 is a new parameter accounting for the importance of the role of parents in the education of their children, while 𝜃 > 0 represents the weight given to leisure time when old. In order to evaluate the stability of the balanced growth path as determined in Proposition 4, in the following we assume 𝜏= 1 − 𝛿 Fig. 2. Trajectories over time of 𝜇𝑡 and 𝑔𝑡 for the initial condition 𝑘0= 0.1, ℎ0= 0.1, 𝑣0= 0.1, 𝜇0= 0.5. since we are mainly interested in the balanced growth path. This is because under such an assumption the models become more tractable and some results may be reached.5 Furthermore, as parameter 𝜃 > 0 only affects quantitative dynamics of the evolution of 𝜇𝑡 without affecting its qualitative properties, we set 𝜃= 0.2. In the following simulations, we use different colors to distinguish between different regions associated to the dynamical model. We depict in green the dynamics occurring in region 𝐶 and described by system 𝑇𝐶 as defined in (34). Such a case is related to poor economies with subsistence wages. The red color is associated to the scenario associated to economies with an intermediate level of development and wage income as described by system 𝑇𝐵 in (33), while, finally, economies characterized by a high level of development and wage income, i.e., described by 𝑇𝐴 in (32) are depicted using blue. We first consider the case of economies starting at a very low level of development, i.e. those economies in which the effective wage is at the minimum living value, 𝑤𝑚ℎ𝑚=𝜋√𝑀 𝜙𝑧 corresponding to combinations between physical and human capital per capita low enough, i.e. such that 𝑘𝛼 𝑡ℎ1−𝛼 𝑡≤𝜋√𝑀 𝜙𝑧𝐴(1 − 𝛼). In Fig. 2 we plot the trajectories of 𝜇𝑡 as defined in (34) and 𝑔𝑡 given by (40) starting from a point in region 𝐶. In such a case, the time devoted by grandparents to educating grandchildren converges over time to a constant value (around 0.7029) and also the ratio between human capital growth rate and physical capital growth rate converges to a constant value (around 0.4732). Notice that such a situation does not represent the convergence to a balanced growth path. In fact, as it has been previously underlined, the growth rate of the physical capital per capita is equal to one (as 𝑘𝑡 is constant ∀𝑡) hence, in this case, 𝑔𝑡 corresponds to the human capital growth rate and it converges to a level that is less than one providing that, the human capital level decreases over time. We now move to the case of economies starting at an intermediate level of development, i.e. those economies in which the effective wage is at intermediate values, corresponding to combinations between physical and human capital per capita low enough, i.e. such that 𝜋√𝑀 𝜙𝑧𝐴(1 − 𝛼)< 𝑘𝛼 𝑡ℎ1−𝛼 𝑡≤𝜋𝑀 𝜙𝑧𝐴(1 − 𝛼). In Fig. 3 we plot the trajectories of 𝜇𝑡 as defined in (33) and 𝑔𝑡 starting from a point in region 𝐵. A transition from region 𝐵 to region 5We underline that the more general case with 𝜏≠1 − 𝛿 can be described in terms of numerical experiments. We have done numerous experiments and have observed that the findings are not different from those obtained by 𝜏= 1 − 𝛿. The Journal of the Economics of Ageing 32 (2025) 100598 7 M.M. Baldi et al. Table 1 Parameter ranges with their most common values. Parameter Range Particular values 𝛼0< 𝛼 < 1𝛼= 0.25 in Goldberger (1968) 𝛼=1 3 in De La Croix and Doepke (2003) and Coppier et al. (2021) 𝐴 𝐴 > 0𝐴= 1.01 in Goldberger (1968) 𝐴= 60 in Coppier et al. (2021) 𝛽0< 𝛽 < 1𝛽= 0.1 in Coppier et al. (2021) 𝛽= 0.169 in De La Croix and Doepke (2004) 𝛿0< 𝛿 < 1𝛿= 0.6 in De La Croix and Doepke (2004) 𝛿= 0.7 in Coppier et al. (2021) 𝜙0< 𝜙 ≤1𝜙= 1 in Yakita (2010) and De La Croix and Doepke (2004) 𝜆0< 𝜆 < 1𝜆= 0.1 in Hirazawa and Yakita (2009) 𝜆= 0.3 in Coppier et al. (2021) 𝜋 𝜋 > 0𝜋= 0.2 in De La Croix and Licandro (2013) 𝜋= 1 in Cipriani and Fioroni (2019) 𝜌0< 𝜌 < 1𝜌= 0.1 in Coppier et al. (2021) 𝜌= 0.99120 in De La Croix and Doepke (2003) 𝑧0< 𝑧 < 1𝑧= 0.075 in De La Croix and Doepke (2004) 𝑧= 0.1 in Coppier et al. (2021) Fig. 3. Trajectories over time of 𝜇𝑡 and 𝑔𝑡 for the initial condition 𝑘0= 10, ℎ0= 10, 𝑣0= 10, 𝜇0= 0.5. 𝐶 can be observed, providing that the economy moves toward less developed situations in which, similarly to what occurred in the case previously described, the time devoted by grandparents to educating grandchildren converges over time to a constant value (around 0.7073) and also the ratio between human capital growth rate and physical capital growth rate converges to a constant value (around 0.48). After a high number of simulations starting from region 𝐵, we observe that such a region is not positively invariant, that is trajectories starting from 𝐵 will exit 𝐵 producing trajectories converging to 𝐶. We consider the case of developed economies, i.e., those economies in which the effective wage is at sufficiently high values, corresponding to combinations between physical and human capital per capita high enough, i.e. such that 𝑘𝛼 𝑡ℎ1−𝛼 𝑡>𝜋𝑀 𝜙𝑧𝐴(1 − 𝛼). In this case, several numerical simulations show that two different situations may emerge. On the one hand, we observe that there are initial conditions producing trajectories entering in region 𝐶, and, hence, converging to a constant equilibrium level for both 𝜇𝑡 and 𝑔𝑡 in the long run. This case is in Fig. 4. Then the asymptotic dynamics converge to a situation in which the physical capital per capita level remains constant over time, while the human capital per capita growth rate converges to a given level less than one (around 0.4732), while the time fraction devoted by Fig. 4. Trajectories over time of 𝜇𝑡 and 𝑔𝑡 for the initial condition 𝑘0= 100, ℎ0= 100, 𝑣0= 100, 𝜇0= 0.5. grandparents to educating grandchildren converges to a constant level (around 0.7029). On the other hand, we observe that there are initial conditions producing trajectories remaining in region 𝐴. This is depicted in Fig. 5, in which case the time devoted by grandparents to educating grandchildren converges to one, while the ratio between human capital per capita growth rate and physical capital per capita growth rate increases over time, following a U-shaped pattern. In this case, the role played by grandparents becomes crucial, as they dedicate all their time to educating their grandchildren. This results in the growth rate of human capital being significantly higher than that of physical capital, leading to paths with unbalanced growth rates. In fact, in economies with a high concentration of investment in education, human capital can grow faster than physical capital. Therefore, the accumulation of human capital, if not properly balanced, can produce divergent trajectories from the accumulation of physical capital. According to the numerical experiments presented, we observe that asymptotic dynamics may occur only in region 𝐴, characterizing developed economies or in region 𝐶, referring to non-developed economies. In the first case, both human and physical capital per capita increase over time, being their ratio greater than one, while grandparents tend to devote all their time to grandchildren’s education. We refer to such a case as a best scenario. In the second case, both human and physical The Journal of the Economics of Ageing 32 (2025) 100598 8 M.M. Baldi et al. Fig. 5. Trajectories over time of 𝜇𝑡 and 𝑔𝑡 for the initial condition 𝑘0= 130, ℎ0= 130, 𝑣0= 130, 𝜇0= 0.5. capital per capita converge to a constant growth rate while grandparents devote only a fraction of their time to educating grandchildren (worst scenario). Hence, we observe that different initial states produce trajectories converging to different asymptotic scenarios, and the sets of such conditions differ from region 𝐶 and 𝐴. To better investigate such a situation, we need to discuss the basins of attraction of the best and worst scenarios. Condition 𝑓1(ℎ𝑡, 𝑘𝑡) = 𝑘𝛼 𝑡ℎ1−𝛼 𝑡−𝜋√𝑀 𝜙𝑧𝐴(1 − 𝛼) defines a curve in the plane R2 + such that points below 𝑓1 correspond to very low developed economies, i.e., those described by the system 𝑇𝐶 as defined in (34). Condition 𝑓2(ℎ𝑡, 𝑘𝑡) = 𝑘𝛼 𝑡ℎ1−𝛼 𝑡−𝜋𝑀 𝜙𝑧𝐴(1 − 𝛼) defines a curve in the plane R2 + such that points above 𝑓1 and below 𝑓2 correspond to economies at an intermediate level of development, i.e. those described by system 𝑇𝐵 as defined in (33), while points above 𝑓2 described developed economies whose dynamics are defined by system 𝑇𝐴 in (32). In Fig. 6, we present in green the basin of attraction of the worst scenario and in blue the basin of attraction of the best scenario, together with the two curves 𝑓1 and 𝑓2. As can be observed, all economies starting from a low level of development, i.e. below the curve 𝑓2, will converge to the worst scenario, while not all economies starting from a high level of development will converge to the best scenario. Following De La Croix and Licandro (2013), we can refer to the green region above curve 𝑓2 as a subsistence trap. The curve separating the green and blue region seems to be well approximated by a hyperbole, which means that there is a given level of human capital, namely  ℎ such that the economy will fall into the subsistence trap for all initial levels of capital per capita as long as the initial level of human capital per capita ℎ0 is less than the threshold  ℎ. The results previously presented depend on the choices of the parameters. Since the main goal of the present work is to consider the influence of the attitude of grandparents in spending time educating grandchildren, we present a diagram showing the effect on the final dynamics related to different selfishness levels of grandparents measured by 𝜃. In Fig. 7, we present the bifurcation diagram w.r.t. 𝜃 showing the long-run equilibrium level of the fraction of time devoted to educating grandchildren, denoted by 𝜇∗, and that of the ratio between the human capital per capita and the physical capital per capita growth rate, denoted by 𝑔∗. It should be noted that if 𝜃 is low, the corresponding 𝑔∗ value cannot be shown, since, as previously discussed, it is not converging as it increases over time. In addition, as the initial state Fig. 6. Basins of attraction of the best (blue) and worst (green) scenarios 𝐴 and 𝐶 in the plane (ℎ0, 𝑘0) being 𝑣0=ℎ0 and 𝜇0= 0.5. Fig. 7. Bifurcation diagram w.r.t. 𝜃 being 𝑣0=ℎ0=𝑘0= 130 and 𝜇0= 0.5. variables are fixed at the same level, a system converging to the best scenario for low values of 𝜃 then moves to the worst scenario as long as 𝜃 assumes higher values. This is because the selfishness of grandparents, i.e. the importance of their free time when they are elderly, decreases the long-run equilibrium level of the fraction of time devoted to educating grandchildren. Reducing the time grandparents devote to educating their grandchildren has a negative impact on human capital and thus on growth, bringing the initially developed economy into the worst scenario. Finally, we underline that, even if we made our experiments in the particular case 𝜏= 1 − 𝛿, we observed that also by choosing different values of 𝜏, the main qualitative results do not change, i.e. region 𝐵 is not attracting, while the final outcome of the system may occur in region 𝐶 or in region 𝐴, depending on the initial conditions. In particular, the system will end in a situation in which grandparents devote all their time to educating grandchildren, and the human capital per capita growth rate is greater than the physical capital per capita growth rate, only if the economy starts from a sufficiently high level of human capital. In Fig. 8 we present the basins of attraction associated with the case in which 𝜏 is less (panel (𝑎)) or greater (panel (𝑏)) than 1 − 𝛿. One can be observe the minimum threshold level of human capital per capita ℎ∗ representing a necessary condition to avoid the subsistence trap from decreasing w.r.t. 𝜏. As 𝜏 increases, i.e. the productivity of parental educational expenditure in the production of human capital, an equal amount of educational expenditure by parents has a greater impact in the formation of their children’s human capital and thus increases the basin of attraction of the best scenario (in blue). Therefore, in such a case, a developed economy is more likely to remain so and not end up in the worst scenario (in green). The Journal of the Economics of Ageing 32 (2025) 100598 9