Improving the quality of life and longevity of the elderly: the role of private versus public health

ORIGINAL RESEARCH Received: 29 April 2025 / Accepted: 21 October 2025 © The Author(s) 2025 Mauro Maria Baldi [email protected] Raffaella Coppier [email protected] Elisabetta Michetti [email protected] 1 Department of Economics and Law, University of Macerata, Macerata, Italy Improving the quality of life and longevity of the elderly: the role of private versus public health Mauro MariaBaldi1· RaffaellaCoppier1· ElisabettaMichetti1 Annals of Operations Research https://doi.org/10.1007/s10479-025-06920-1 Abstract We develop an overlapping generations model to study how the combination of public and private health expenditures affects the health status and/or longevity of the elderly, as well as its impact on steady-state economic growth. We find that two distinct scenarios may arise-one with and one without reliance on private health care-depending on the relative value of private versus public health spending. In both cases, a positive locally asymptotically steady state emerges in terms of capital per worker, and a switch between regimes may occur depending on the share of public balance spent on the health system. Furthermore, increasing such a share increases the equilibrium longevity, while the effects on health status are ambiguous. Specifically, when the effectiveness of public expenditure is low, increasing public resources allocated to healthcare does not necessarily lead to improvements in health status. In contrast, when public spending is highly effective, greater allocation of public resources becomes a powerful tool to improve health status in old age. Keywords Public and private health expenditures · Longevity · Life quality of the elderly · Economic growth · Overlapping generations model 1 Introduction In recent decades, nearly all countries have experienced a substantial increase in human longevity. Life expectancy estimates the average number of years a person is expected to live based on various factors such as their birth year, current age, and demographic characteristics. It is a key indicator used to assess the general health and well-being of populations, often reflecting the effectiveness of healthcare systems, socioeconomic conditions, lifestyle choices, 1 3 Annals of Operations Research and public health policies. Between 1990 and 2022, life expectancy improved in all regions of the world, leading to a reduction in disparities between countries. In 2022, Chad (52.997 years), Lesotho (53.036 years), and Nigeria (53.633 years) had the lowest life expectancy at birth (LEB), while Japan (84.82 years), Liechtenstein (84.656 years) and Switzerland (84.255 years) reported the highest LEB. In 1990, the difference between the highest and lowest LEBJapan and South Sudan, respectively-was 49.049 years (Aytemiz et al., 2024). By 2022, this gap had narrowed to 31.823 years between Japan and Chad. Despite this progress, the remaining disparity is still substantial, highlighting the need to focus on achieving the Sustainable Development Goals (SDGs) set forth in the 2030 Agenda for Sustainable Development. (United Nations, 2024)1 Longevity has been a cornerstone in public health and medical debates and life expectancy is often used to measure the health and well-being of a population (Annas & Galea, 2018). At least in the developed world, longer expected lifespans have been accompanied by a significant increase in healthcare expenditures. A growing body of literature shows a clear link between health expenditures and health outcomes (Glied & Smith, 2013). In particular, data from advanced countries show that public health spending plays a very important role in enhancing longevity, a much greater role than private health spending. This evidence stems from the fact that: “public health expenditure is devoted, first and foremost, to finance actions that affect an important fraction of the population and involve significant positive external effects (i.e., what we could identify as basic health: vaccination campaigns, prevention of disease, basic framework of health centers, etc.). But once basic programs are met, additional public expenditure is likely to be devoted to activities that the private sector also offers, so the productivities of the two sectors converge." Aísa et al. (2014). In fact, there are some countries, e.g. the United States, where healthcare expenditure is high, but the public component of this expenditure is low, with limited effects on improving longevity. With regard to this, in 2023, the life expectancy in the United States was 78.4 years, which is 4.1 years lower than the average of comparable countries at 82.5 years. As Aísa et al. (2014) states: “In particular, the above results offer a plausible explanation for the apparently paradoxical data for the USA. While this is the country that devotes the largest amount of resources to health (over 13 % of GDP, close to twice the average value in the sample), life expectancy was 76.2 years in 2000, below the average. The key element that enables us to understand this puzzle is precisely the composition of health expenditure. The average value in the sample shows a ratio of public to private expenditures of 3.72. In contrast, this ratio is only 0.82 for the USA. That is, the public health system in the USA is responsible for only 45 % of the resources devoted to health, compared to the almost 80 % average in the OECD countries. The above results show the need for a redesign of the health system through an intensive promotion of the public health system." Although life expectancy has generally increased, healthy life expectancy (HALE) has not progressed at the same rate.2 Between 2000 and 2019, HALE increased by 5.3 years, from 58.1 to 63.5 years, while overall life expectancy grew by 6.4 years during the same period. This indicates that, although people are living longer, they are also spending more 1 Available at: https://sdgs.un.org/goals.. 2 Healthy life expectancy is a measure that combines both quantity of life and health status. This refers to the average number of years that a person can expect to live in full health, free from disease or disability. It is not just about how long people live, but how long they live without major health problems. 1 3 Annals of Operations Research of those additional years dealing with health problems.3 But, as Zarulli and Caswell (2024) stress, surviving in good health is essential because healthy years of life are also a crucial part of the dynamics of the life cycle. These trends underscore the importance of addressing not only longevity, but also health status during extended lifespans. Efforts to improve access to healthcare, preventive measures, and health education are crucial in ensuring that increased life expectancy translates into healthier years lived. Governments around the world recognize the importance of the health system; therefore, health expenditures throughout the world have increased with time.4 Health expenditures are mostly financed through public taxation and are growing more than the global economy, accounting for 10% of the world gross domestic product (GDP) (25). The average health expenditures as a share of GDP have increased from 7.8% in 2005 to 9.8% in 2020 in the OECD countries (Anwar et al., 2023). In our paper, we propose an economic framework able to account for both elements relevant to health: one related to life expectancy (i.e., life quantity), and one expressing quality from the point of view of the elderly person’s health (i.e., health status). We therefore construct a 2-period overlapping-generations model that incorporates an analysis of endogenous longevity together with the endogenous elderly health status. Therefore, we combine two strands of research by considering both the ‘quantity’ of life and health status. Following (Chakraborty, 2004) and Cipriani and Fioroni (2019), we assume that the probability of survival from the first period (adulthood) to the next (old age) is endogenously determined through public investment in health.5 It is also important that elderly survivors increase their health status. Hence, we assume, following (Varvarigos & Zakaria, 2013), that this state of health depends on both public expenditures on medical care, and also on private spending by individuals during the final period of their lives. In fact, the elderly can decide to allocate part of their income to health care expenditures to improve their utility-enhancing health status. Understanding how public and private health expenditures can interact is essential for shaping or reforming health care policies. Bhattacharya and Qiao (2007) and Varvarigos and Zakaria (2013) consider that private and public health expenditures are complementary, i.e., an increase in one leads to an increase in the other, rather than substituting for it. This occurs when private and public investments reinforce each other in improving elderly health.Unlike(Varvarigos & Zakaria, 2013), we interpret the public component as a "competitor" to the private input in determining elderly health because public and private expenditures have a certain degree of substitutability. The choice to consider both private and public expenditure by the elderly arises from the fact that private expenditure in this context - such as medical treatments and interventions 3 GHE: Life expectancy and healthy life expectancy. The Global Health Observatory. h t t p s : / / w w w . w h o . i n t / d a t a / g h o / d a t a / t h e m e s / m o r t a l i t y - a n d - g l o b a l - h e a l t h - e s t i m a t e s / g h e - l i f e - e x p e c t a n c y - a n d - h e a l t h y - l i f e - e x p e c t a n c y - World Health Organization (2022). 4 WHO. Countries are Spending More on Health, But People are Still Paying Too Much Out of Their Own Pockets. WHO (2019). Available online at: h t t p s : / / w w w . w h o . i n t / n e w s / i t e m / 2 0 - 0 2 - 2 0 1 9 - c o u n t r i e s - a r e - s p e n d i n g - m o r e - o n - h e a l t h - b u t - p e o p l e - a r e - s t i l l - p a y i n g - t o o - m u c h - o u t - o f - t h e i r - o w n - p o c k e t s . 5 The decision not to consider private spending as a determinant of longevity is based on the fact that data on developed countries, such as the United States, as highlighted in the introduction, show that public healthcare spending is the main determinant of a country’s longevity. Furthermore, in our model, longevity is the probability of surviving into the second period, and therefore private spending on longevity should be done by young adults, which is not very common. Thus, this modelling makes the analytical model more manageable without implying a loss of generality of the model itself. 1 3 Annals of Operations Research for existing diseases and conditions - tends to improve the health status of the elderly. These expenditures are primarily focused on care, as prevention is often not an immediate priority for private individuals who allocate their resources mainly toward treatments and therapies. As a result, this type of healthcare can be provided interchangeably by the public health system or, in cases of deficiencies or excessively long waiting times, by the private system. In this regard, (Li et al., 2016) conduct a quantitative exercise to examine whether the observed differences in the public-private mix of health expenditure can be accounted for by variations in the elasticity of substitution between private and public spending, as well as by differences in the effectiveness of public expenditure in health production across a sample of OECD countries. Following the quantitative analysis by Li et al. (2016), our model considers two different scenarios regarding the effectiveness of public health expenditure, each representing two markedly different models of private-public health expenditure mixes. This paper presents a simple framework to study the dynamic substitutability between public health programs and private efforts, aimed at improving health status in old age, given that longevity depends on the public health system. By combining analytical methods and numerical experiments, our model shows that capital per young worker evolves over time according to different dynamics. Specifically, two scenarios may emerge: in the first the value of private health spending is higher than public spending, leading the elderly to allocate part of their savings to health; in the second the value of private health spending is lower than public spending, and the elderly choose not to invest in health. In both cases, the model admits a unique, positive, and locally asymptotically stable steady state. However, as the share of public resources allocated to health care changes, a switch between the two scenarios may occur. While this transition does not affect the equilibrium level of longevity, its impact on the health status may be ambiguous. The paper proceeds as follows. Section 2 illustrates the set up of the model. Section 3 presents the dynamics of the system. Section 4 concludes. 2 The economic setup 2.1 Health status and quantity in the elderly We consider a production economy populated by overlapping generations of agents who live for two periods: adulthood and old age. Following (Chakraborty, 2004), longevity is endogenous, i.e. the probability of surviving from adulthood to old age depends on public expenditures on health. Hence, we assume that longevity depends only on public health expenditures made by the government at time t∈N , and the more the government invests in public healthcare, the greater the probability of an adult surviving as an elderly person. We define as wt the wage rate of an adult for a unit of effective labor, then, being τ∈(0,1) the taxation rate, the public health financed by the state positively depends on fiscal revenues i.e., τwt . We denote by gt the amount of public expenditure on health at time t, then, let γ∈(0,1) be the fraction of the public balance devoted to the health system, we obtain gt=γτwt. (1) 1 3 Annals of Operations Research Although the length of each period is normalized to one, an individual’s lifetime is uncertain. More precisely, the individual probability of surviving from the first period (adulthood) to the next (old age) depends on public expenditures on health when young. We refer to this component as quantity of life in the elderly, and it is described by a variable measuring the survival probability of an adult, thus becoming old in the second period. We denote this probability as pt∈[¯p, 1) , given a fixed exogenous threshold for longevity, which is associated with no investment in public health, denoted by ¯p∈[0,1) . Then, following (Chakraborty, 2004) and Cipriani and Fioroni (2019), we consider a function P:R+→[¯p, 1) which associates the longevity pt with public health expenditure gt , such that, P(0) = ¯p . Specifically, there exists an exogenous level of longevity where limgt→+∞P(gt)→1 , meaning that if public health expenditure is sufficiently high, then longevity is close to one. The function P is continuous, twice differentiable and such that P′>0 , indicating that increased public expenditure on health improves longevity, and P′′ <0 , reflecting diminishing marginal benefits. A function satisfying these assumptions is as follows: p t=¯p+ (1 −¯p)g t 1+gt . (2) Regarding the health status in the elderly, following (Varvarigos & Zakaria, 2013), we assume it is related to the expenditures in the health system through two components: the public one, financed by the state via taxation, and the private one, funded by the elderly themselves, who divert a portion of their resources from consumption in old age. However, unlike (Varvarigos & Zakaria, 2013), we consider private and public health expenditures to act as a perfect substitute to improve the health status of the elderly. The substitutability between private and public health expenditures can be explained by the estimates of Li et al. (2016), which suggest that for most OECD countries, either the Cobb-Douglas form or a linear form is a reasonable representation for health technology. In fact, in old age individuals primarily allocates resources toward treatments and therapies aimed at managing the health effects of existing conditions. In this context, private spending may be a solution to the excessive waiting times in public healthcare and congestion problems related to public spending. We denote private health expenditure in old age as xt+1 , which represents a fraction of the individual’s savings. Thus, these savings are used both for consumption in old age and for improving health status. Regarding the health status of the elderly, it is assumed that the health technology that determines the health status depends on both private and public health expenditures at time t+1 , with a perfect degree of substitutability between them.6 The health status in old age, denoted by ht+1 , can be formalized as follows: ht+1 =αgt+1 + (1 −α)xt+1. (3) 6 Considering imperfect substitutability between public and private spending on health technology (considering, for example, a Cobb Douglas function) implies that the technical substitution rate between the two types of spending is not constant but decreases as the use of one of the two types of spending increases. As we already mentioned, based on the empirical analysis of Li et al. (2016), both types of function (linear or Cobb Douglas) are supported by empirical data, but the use of a Cobb Douglas function would make the theoretical analysis much more complex. 1 3 Annals of Operations Research where α∈(0,1) represents the effectiveness of public expenditure in improving the health status of the elderly, while (1 −α) is the effectiveness of private expenditure by the elderly themselves. The ratio α (1−α) measures the rate at which private and public health expenditure can be exchanged, while maintaining a constant level of health. 2.2 Intertemporal constrained utility maximization We consider an economy consisting of an infinite sequence of overlapping generations, each potentially living for two periods, alongside an infinitely-lived government. Time is discrete, that is, t=1,2, ... . Following (Chakraborty, 2004), we assume that each individual born in generation t gives birth to one offspring at the end of period t, before experiencing their mortality shock. The new individual becomes economically active only at the beginning of t+1 . Therefore, in each period a measure-one cohort of adults is born, each endowed with one unit of time, which they inelastically supply to the labor market, earning a wage income wt . Following (Cipriani & Fioroni, 2019), to keep the model more tractable, we assume that all adults retire at the end of the first period. In summary, adults benefit from the consumptions during adulthood, consumptions in old age, and their health status ht+1 when old, depending on the survival probability. The lifetime utility of an individual of generation t is then given by the following function7: Ut= ln ct+pt{βln ct+1 +θln ht+1}, (4) where ct is consumption during adulthood, ct+1 is consumption during old age, and ht+1 is the health status during old age. The parameters β , θ are positive, and pt represents the probability of surviving from youth to old age. The budget constraint in the first young adulthood period is given by: (1 −τ)wt=ct+st (5) where τ∈(0,1) is the tax rate on labor income and st refers to savings (understood as the purchase of annuities). All variables are assumed to be non-negative. Given the adults’ salary and the interest rate, the cost of health expenditure reduces the resources available for both future consumption and savings. The second-period budget constraint is given by: stˆ Rt+1 =ct+1 +xt+1 (6) where xt+1 is the private expenditure for health status, and ˆ Rt+1 is the gross return on its savings. In fact, consumption in old age is financed by the returns on savings accumulated during adulthood. Furthermore, we assume that all goods are perishable and that agents can 7 The assumption of additivity in the utility function in overlapping generations models is often used (see, for example (De La Croix & Michel, 2002)). Additivity implies that the well-being derived from consumption does not depend on the level of health and vice versa. In reality, consumption is often more “useful” if you are healthy and vice versa: in fact, greater consumption can improve health status. However, this hypothesis allows us to analyze choices relating to consumption and health separately: in fact, the agent decides to invest part of their savings in health only by looking at the direct contribution to utility, without considering how health affects the marginal utility of consumption. 1 3 Annals of Operations Research only transfer value over time through capital markets. Individuals are assumed to have no bequest motives. Following (Chakraborty, 2004), in order to eliminate the risks associated with uncertain lifespans, we assume the existence of a perfect annuity market, where all savings are managed through mutual funds. At the end of their youth, individuals deposit their savings into a mutual fund. These funds are exclusively invested in capital, and the mutual fund guarantees a gross return to those who survive into old age. If these funds yield a gross return of ˆ Rt+1 on its investment, then under perfect competition, equilibrium in the annuity market is maintained: ˆ R t+1 = R t+1 pt , (7) where Rt+1 is the gross interest rate. To summarize, we can collect all the equations presented so far to formulate the following constrained optimization problem: max Ut= ln ct+βptln ct+1 +θptln ht+1 (8) s.t.: gt+1 =γτwt+1 (9) ct+st=(1 −τ)wt (10) ct+1 +xt+1 =stˆ Rt+1 (11) ˆ R t+1 = R t+1 pt (12) ht+1 =αgt+1 + (1 −α)xt+1 (13) xt+1 ≥0. (14) The solution of the maximum constrained optimization problem results in the following Proposition. Proposition 2.1 The first-order conditions for problem (8–14) are as follows. Case A: If (1 −α)θRt+1(1 −τ)wt≥(1+βpt)αγτwt+1 (15) then c t= 1 1+( β + θ ) p t[ pt R t+1 α 1− αγτwt+1 + (1 − τ)wt ], (16) 1 3 Annals of Operations Research c t+1 =βRt+1 1+(β+θ)pt [ pt Rt +1 α 1 − αγτwt+1 + (1 −τ)wt ], (17) x t+1 = 1 1+(β+θ)pt[ θRt+1(1 − τ)wt − (1+βpt) α 1−α γτwt+1 ], (18) s t=pt 1+(β+θ)pt [ (β+θ)(1 −τ)wt− 1 Rt +1 α 1 − αγτwt+1 ]. (19) Case B: If (1 −α)θRt+1(1 −τ)wt<(1+βpt)αγτwt+1 (20) then c t= (1 −τ)w t 1+βpt , (21) c t+1 = βR t+1 (1 −τ)w t 1+βpt , (22) xt+1 =0, (23) s t=βpt (1 − τ ) wt 1+βpt . (24) Moreover, in both cases, these conditions are also sufficient. Proof See Appendix A. □ The previous conditions referred to Case A and Case B become easier to interpret economically. In fact, the condition relating to Case A is: (1 −α)θRt+1(1 −τ)wt≥(1+βpt)αgt+1. (25) The left-hand side of the inequality represents the value ( θ ) of the (net, discounted) income spent on private expenditure when elderly (also taking into account the effectiveness 1−α ), while the right-hand side, on the other hand, represents the value of public expenditure gt+1 of which the effectiveness α is also taken into account. Therefore, when the inequality of Case A is verified, it means that the value of private health expenditure is greater than public expenditure. As a consequence, private spending on health by the elderly is positive. Conversely, in Case B the value of private spending on health is lower than public spending, and therefore the elderly will not invest their savings in health spending, i. e. xt+1 =0 . Thanks to the first-order conditions, we can also derive ht+1 by substituting the expression of xt+1 in both cases A and B (respectively given by (18) and by (23)) into the formula (3) and considering that gt+1 =γτwt+1 . After some algebraic manipulations, the expression for ht+1 in Case A is as follows: 1 3 Annals of Operations Research h A t+1 = [ αγτwt+1pt + (1 − α )(1 − τ ) Rt+1wt ] θ 1+(β+θ)pt , (26) while for Case B, we obtain: hB t+1 = αγτw t+1. (27) It is straightforward to observe that hA t+1 ≥ h B t+1 . In fact, from (13), (9), and (14) we have: ht+1 = αg t+1 + (1 − α ) x t+1 ≥ αg t+1 = αγτw t+1 = h B t+1. (28) In particular, (28) holds if hA t+1 is substituted on the left-hand side. Alternatively, one arrives at a similar conclusion by bounding (26) using (15). In fact, from (15) we have: h A t+1 = αγτw t+1 p t θ+ (1 −α)(1 −τ)R t+1 w t θ 1+(β+θ)pt ≥ αγτwt+1ptθ+(1+βpt)αγτwt+1 1+(β+θ)pt =αγτwt+1 =hB t+1 . As for the expression of pt , in both cases we find that by substituting (1) into (2), we obtain: p t=¯p+ (1 −¯p)g t 1+gt =¯p+ (1 −¯p)γτw t 1+γτwt . (29) Thus, it can be seen that the quantity of life, i.e. longevity, is equal in the two scenarios as it depends only on public spending; conversely, health status is higher in scenario A where the elderly invest part of their savings in improving their health. 2.3 Production, investment and saving As previously mentioned, at each period, a new generation of adults, each with measure one, enters the economy. Each agent is endowed with one unit of labor during their youth and is compulsorily retired in old age. The aggregate production technology of the economy is assumed to follow a constantreturns-to-scale production function, utilizing both labor and capital. Capital stock is assumed to fully depreciate after one period of use, meaning that the capital stock in any given period is equal to the savings in the previous period. Production in time t employs physical capital Kt and labor L. We denote Yt the aggregate output and represent the aggregate technology of the economy by the following production function: Yt= F ( K t ,L )= AK δ t L 1−δ. (30) In that equation, Yt , Kt , and L respectively stand for aggregate output, physical capital stock and effective labor in the economy in period t, while A is the total factor productivity, and δ∈(0,1) is the productivity of physical capital. 1 3 Annals of Operations Research A key question, however, concerns how the system transitions between these scenarios as a parameter -specifically, the share of the public budget allocated to the healthcare systemchanges. We first consider the case with low α -value, i.e. α=0.5 so that, as emerges from Fig. 1b, the fixed point always belong to RA , i.e., a positive fraction of savings is allocated to private healthcare for all levels of public budget devoted to health system γ . The resulting equilibrium value of capital per worker, k∗=k∗ A as moving γ∈(0,1) , is shown in Fig. 4a. The virtual fixed point associated with the no-private-spending scenario typically results in a lower level of k∗ , assuming the feasible equilibrium solution is maximizing. However the corresponding curve (in cyan) exhibits a hump-shaped pattern: as γ increases, the equilibrium value of capital per worker initially rises and then declines. This suggests the existence of an optimal public budget share for healthcare that maximizes the equilibrium outcome. We now turn to the case of a high α -value, specifically α=0.8 . In contrast to the previous case, the resulting scenario here depends on the γ -value, the share of the public budget allocated to the health system. For low values of γ , the model again produces scenario A, characterized by positive healthcare spending. In this case, the maximum capital per young worker level is reached at γ≃0.11 . However, as the share of the public spending on healthcare increases and crosses the threshold value γ≃0.6 , a transition occurs: the fixed point k∗ A becomes virtual, while the fixed point k∗ B becomes feasible. This marks a shift to a new situation in which there is no private healthcare spending. In this case, since public healthcare spending is highly effective, then there exists a threshold level of tax revenue allocated to healthcare that shifts the equilibrium from Case A to Case B. This implies that, if public spending is highly productive in generating “health", there is no need for the elderly to allocate part of their savings to health expenditure. In these countries, an effective and extensive public healthcare system alone is sufficient to ensure a good health status. Conversely, if public and private healthcare spending are equally effective, only the scenario involving private health expenditure can emerge. In this case, no level of public spending is sufficient to prevent the elderly from investing in “health". Finally, we aim to asses whether, and to what extent, the optimal choices emerging from both scenarios affect both life quantity and health status. Fig. 4 The equilibrium point k∗ for region A (in cyan) and region B (in magenta) being γ∈(0,1) 1 3 Annals of Operations Research Longevity is given by pt(kt) as defined in equation (35), while health status in old age -depending on the scenariois described by the following equations: h A t+1 = { αγτkδ t+1 [ ¯p+ (1 − ¯ p ) γτA (1 −δ ) k δ t 1+γτA(1−δ)kδ t ] + (1 −α)(1 −τ) ( 1+Aδkδ− 1 t+1 ) kδ t } A(1 −δ)θ 1+(β+θ) [ ¯p+(1−¯p)γτA(1−δ)kδ t 1+γτA(1 − δ)kδ t], (42) in the case of positive private healthcare spending, and hB t+1 = αγτA (1 − δ ) k δ t+1, (43) in the case of no private healthcare spending. With regard to longevity, even if its functional form remains the same across both scenarios, it varies with the equilibrium level of capital per worker, which in turn depends on the share of the public budget γ allocated to the healthcare system. As shown clearly in both Figs. 5a and 6a, life expectancy increases as γ increases, even in the absence of private healthcare spending. This evidence holds regardless of whether public spending is relatively effective or not and it persists even when transitions between scenarios occur. This is because longevity is determined solely by public healthcare spending, not private expenditure, which makes the outcome similar in both scenarios. However, a different behavior emerges when considering health status. Specifically, when the effectiveness of public healthcare spending is low, increasing the share of the public budget dedicated to health does not necessarily improve health statusthis is evident in Fig. 5b. In contrast, when public spending is highly effective, even a shift between scenarios, does not hinder improvements: increasing the public budget share dedicated to health becomes an effective tool for enhancing the equilibrium level of health status (see Fig. 6b). Indeed, when public healthcare spending is significantly more productive than private spending, a scenario with only public healthcare -without any private health expenditure-may result in a higher health status. This provides important policy insights: if public healthcare spending is significantly more effective than private alternatives, and the State Fig. 5 Equilibrium values of p∗ (panel a) and h∗ (panel b) with α=0.5 for region A (in cyan) and region B (in magenta) being γ∈(0,1) 1 3 Annals of Operations Research chooses to allocate a large share of its tax revenue to healthcare, a fully public system can ensure a higher health status than a mixed public-private model. 4 Conclusion We considered an overlapping generations model to investigate the role of public versus private healthcare in determining both the quantity of life and health status in old age, as well as its impact on steady-state economic growth. Depending on the comparison between the value of private and public health expenditures, two scenarios can emerge: either the elderly allocate a positive amount of their savings to private healthcare, or they choose not to invest in private health at all. With regard to the existence and stability of steady-state equilibria in capital per worker - and the corresponding dynamics of life quantity and health status - we combine analytical tools with numerical methods to demonstrate that, in each scenario, a positive, locally asymptotically stable steady-state emerges. This steady state can be either feasible or virtual, depending on the model’s parameter values. Accordingly, we fixed the values of the main parameters and vary both the share of the public budget allocated to healthcare and the efficiency of public spending in the health production function. On the one hand, our work is able to show that if public and private spending are equally effective in the health production function, then the feasible steady-state is characterized by positive private health expenditure by the elderly, regardless of the share of the public budget allocated to the health system. Differently, if public spending is more efficient than private expenditure, then there is a transition from a situation with positive private healthcare spending to a one with no private healthcare spending, as long as the share of the public budget devoted to the healthcare system is increased. Fig. 6 Equilibrium values of p∗ (panel a) and h∗ (panel b) with α=0.8 for region A (in cyan) and region B (in magenta) being γ∈(0,1) 1 3 Annals of Operations Research On the other hand, it shows that an increase in the share of the public budget allocated to the healthcare system has a positive effect on longevity in both scenarios, whether or not the elderly engage in private healthcare spending. However, the impact of this increase on health status may be ambiguous, as it depends critically on the effectiveness of public spending within the health production function. Specifically, when public expenditure is relatively ineffective, increasing the allocation of public resources to healthcare does not necessarily result in improvements in health status. In contrast, when public spending is highly effective, a larger allocation of public resources becomes a powerful tool to improve health status in old age. The abovementioned results results suggest that policymakers could improve the effectiveness of public health expenditures by prioritizing preventive care programs, improving access to primary healthcare services, and investing in vaccination and disease control initiatives. In addition, targeted subsidies for essential treatments and the expansion of public hospital capacity could help ensure that increased public spending is translated into tangible improvements in both longevity and health outcomes. 4.1 Further developments of the model can be considered in future research First, different health technologies could be considered that take into account the imperfect substitutability between public and private healthcare expenditures. In fact, considering imperfect substitutability, such as by adopting a Cobb-Douglas specification, the rate of technological substitution between public and private expenditure would no longer remain constant but would diminish. This means, for example, that as public healthcare expenditure increases, its marginal ’productivity’ would fall relative to private expenditure. This adjustment could alter the optimal allocation between public and private spending; however, analyzing this case would require a new model, which may not be analytically tractable given the added complexity introduced by non-linearities. Furthermore, the assumption of additivity in the utility function implies that the utility derived from consumption is independent of the individual’s health status, and vice versa. The additive specification allows consumption and health-related decisions to be analyzed separately. Under this assumption, agents allocate savings to health solely based on its direct contribution to utility, without accounting for the way health might influence the marginal utility of consumption. In contrast, a non-additive (e.g., multiplicative) utility specification introduces complementarity between health and consumption. In such a framework, better health increases the marginal utility of consumption, thereby increasing the likelihood of higher investment in healthcare. Similarly, greater consumption can reinforce health and improve its marginal utility. While this approach is more realistic, it significantly increases the model’s complexity and eliminates the possibility of treating the two choices independently. However, it represents a promising and natural extension, which we plan to pursue in future research. Finally, an important extension of the model in future research could be to consider that longevity, as well as health in old age, depends not only on public spending but also on private spending. This inclusion could perhaps make the model difficult to study analytically, but it would certainly add interesting elements for further reflection. 1 3 Annals of Operations Research Appendix A We begin by making preliminary substitutions to simplify the maximization problem, reducing the number of constraints. To do this, we substitute the expression for gt+1 given by (9) into (13), and the expression for ˆ Rt+1 given by (12) into (11). Then, from (11), we calculate st as s t =p t Rt+1 ( ct+1 + xt+1 ) (A.1) and substitute this expression into (10). Finally, we substitute (13) into (8) and obtain the following equivalent model: max Ut= ln ct+βptln ct+1 +θptln [αγτwt+1 + (1 −α)xt+1] (A.2) s.t.: ct+ p t Rt+1 (ct+1 +xt+1) = (1 − τ)wt (A.3) −xt+1 ≤0. (A.4) We solve model (A.2–A.4) using the method of Lagrange multipliers. In this regard, we define ηt as the Lagrange multiplier associated with constraint (A.3), and ξt as the Lagrange multiplier associated with constraint (A.4). We observe that model (A.2–A.4) is a mixed constrained maximization problem, as it includes both an equality constraint and an inequality constraint. According to, the theorem on Lagrange multipliers (see, for instance, Simon and Blume 1994, pp. 434 and 435), the constraint qualification requires that the rank of the Jacobian matrix -evaluated at a local maximizer and corresponding to the set of equality constraints and binding inequality constraintsbe maximal. In our case, this matrix, regardless of the point at which it is evaluated, is given by: [1p t Rt+1 p t Rt+1 00 − 1 ]. Since its rank is always maximal, the constraint qualification condition is satisfied. We can therefore construct the Lagrangian function, whose expression is: L t (c t ,c t+1 ,η t ,ξ t ) = ln c t +βp t ln c t+1 +θp t ln [αγτw t+1 + (1 −α)x t+1 ] − ηt [ ct+pt Rt+1 (ct+1 +xt+1) − (1 − τ)wt ] +ξtxt+1 By differentiating the Lagrangian with respect to the decision variables ct , ct+1 , and xt+1 we respectively obtain: ∂L t ∂ct =0: 1 ct − ηt =0 (A.5) 1 3 Annals of Operations Research ∂L t ∂ct+1 =0: βp t ct+1 − ηt p t Rt+1 =0 (A.6) ∂ L t ∂xt+1 =0: (1 − α ) θpt αγτwt+1 + (1 −α)xt+1 − ηt pt Rt+1 +ξt =0. (A.7) It is also necessary to satisfy the initial constraints: c t +p t Rt+1 ( ct+1 + xt+1 ) = (1 − τ ) wt (A.8) and xt+1 ≥0. (A.9) Moreover, the complementary slackness condition requires that: ξtxt+1 =0. (A.10) Finally, the non-negativity condition for the multiplier associated with the inequality constraint is given by: ξt≥0. (A.11) From (A.5) we get c t= 1 ηt . (A.12) Likewise, from (A.6) we get c t+1 = βR t+1 ηt . (A.13) We first consider the case when xt+1 >0 . We refer to this case as Case A. In light of (A.10), this implies that ξt=0 . By replacing ξt=0 into (A.7), after some algebra, we find the following expression for xt+1 : x t+1 = θ ηt Rt+1 −α 1−α γτwt+1 . (A.14) By substituting (A.12), (A.13), and (A.14) into (A.3), we find η t= 1 1 1+(β+θ)pt [ pt Rt+1 α 1 − αγτwt+1 + (1 − τ)wt ]. (A.15) 1 3 Annals of Operations Research By substituting this value into (A.12), (A.13), and (A.14), we obtain the value of ct , ct+1 , and xt+1 respectively. Finally, by imposing xt+1 >0 , we get the following condition: (1 −α)θRt+1(1 −τ)wt>(1+βpt)αγτwt+1. (A.16) Case B arises when ξt>0 , which implies xt+1 =0 . From (A.7), we obtain ξ t= p t Rt+1 ηt −(1 −α)θp t αγτwt+1 + (1 −α)xt+1 . (A.17) By substituting (A.12), (A.13), and xt+1 =0 into (A.3), we find η t= 1+βp t (1 −τ)wt . (A.18) By substituting this value into (A.12) and (A.13), we obtain the value of ct and ct+1 , respectively. Moreover, by substituting (A.18) into (A.17), we find the condition characterizing Case B. The case when ξt=0 and xt+1 =0 represents the boundary between the two regions corresponding to Cases A and B. This justifies the use of ≥ (instead of >) in the inequality describing Case A. The expressions (19) and (24) for st are obtained by substituting the values of ct+1 and xt+1 into (A.1), respectively for Cases A and B. Finally, the Hessian matrix of the Lagrangian function with respect to the decision variables ct , ct+1 , and xt+1 is given by: H Lt=    − 1 c2 t 0 0 0−βpt c2 t+1 0 00 − θ(1−α)2pt [αγτwt+1+(1−α)xt+1] 2   . (A.19) This matrix is clearly negative definite. Consequently, the first-order conditions are also sufficient. Appendix B From (31), it follows that wt+1 = A (1 − δ ) k δ t+1. (B.1) Likewise, (32) implies rt+1 = Aδk δ t+1. (B.2) Consequently, from (33) we obtain: 1 3 Annals of Operations Research Rt+1 =1+ r t+1 =1+ Aδk δ t+1. (B.3) From (29) and (31), we can obtain a new expression for pt as follows: ¯ p + (1 − ¯ p ) γτwt 1+γτwt =¯p+ (1 − ¯ p ) γτA (1 −δ ) k δ t 1+γτA(1 −δ)k δ t =: pt(kt) . (B.4) As stated in Proposition 2.1, the boundary separating regions A and B is given by the following equation: θR t+1 (1 − τ ) wt = (1 + βpt )α 1−α γτwt+1 . (B.5) Bringing all terms to the left-hand side, applying the previously mentioned substitutions, and making the dependence of pt on kt explicit as stated in (B.4), we define the left-hand side of this new equation as fC(kt,k t+1) . This allow us to express the boundary condition in terms of the equation [ θ ( 1+Aδkδ− 1 t+1 ) (1 − τ)kδ t − (1+βpt(kt)) α 1−α γτkδ t+1 ] A(1 − δ )=0. Next, for both Cases A and Case B, we substitute the expression for st from (34), taking into account the previous substitutions both in the expression of st and in the conditions characterizing each case. From this, the thesis follows. As with the boundary condition, all terms must be brought to the left-hand side. We then define fA for region A and fB for region B. Appendix C (a) It is trivial to verify that ˆ fB(0) = 0 . (b) By applying the chain rule to compute ˆ f ′ B( k t) , we obtain: ˆ f ′ B(kt)=β (1 − τ ) A (1 − δ ) (1+βp t(kt)) 2 [ p′ t(kt)kδ t+δp t(kt)kδ−1 t(1+βp t(kt)) ]. (C.1) Since all the parameters are positive, kt≥0 , pt(kt)>0 for all kt≥0 , and p′(kt)>0 for all kt≥0 , it follows that ˆ f′ B(kt)≥0 for all kt≥0 . (c) Exploiting the fact that for all kt≥0 it holds that ¯p≤pt(kt)<1 , and we obtain: ˆ f B(kt)> β¯p(1 −τ)A(1 −δ) 1+β kδ t . (C.2) Applying the squeeze theorem, the claim follows. (d) We compute the first derivative at 0 by taking the limit of the incremental ratio as h→0+ . Specifically, also in light of (C.2), we have: 1 3 Annals of Operations Research lim kt→ 0+ ˆ f′ B(kt) = lim h → 0+ ˆ fB ( h ) − ˆ fB (0) h = lim h → 0+ ˆ fB ( h ) h >lim h → 0+ β ¯ p (1 −τ ) A (1 −δ ) 1+β 1 h 1 − δ=+ ∞. Funding This work has been funded by the European Union - NextGenerationEU under the Italian Ministry of University and Research (MUR) National Innovation Ecosystem grant ECS00000041 - VITALITY - CUP D83C22000710005. Declarations Conflict of interest We declare that there are no conflict of interest. Open Access This article is licensed under a Creative Commons Attribution-NonCommercialNoDerivatives 4.0 International License, which permits any non-commercial use, sharing, 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 you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. 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