# Should I Wait or Should I Pay? The Dynamics of Private and Public Healthcare

Denis Caporale, Francesco Magris, Gabrielle Sbaiz, Bianca Tanasa

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Denis Caporale, Francesco Magris, Gabrielle Sbaiz, Bianca Tanasa. Should I Wait or Should I Pay? The Dynamics of Private and Public Healthcare. 2025. hal-05344962

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Document de Recherche du Laboratoire d'Économie d'Orléans

Working Paper Series, Economic Research Department of the University of Orléans, University Of Tours
and University Clermont-Auvergne, France

DR LEO 2025-23

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# Should I Wait or Should I Pay?
# The Dynamics of Private and
# Public Healthcare

Denis Caporale, Francesco Magris, Gabriele Sbaiz et
Bianca Tanasa

MISE EN LIGNE / ONLINE | 03/11/2025

<figure><img src="image_4.png" /></figure>

Laboratoire d'Économie d'Orléans
Collegium DEG
Rue de Blois - BP 26739
45067 ORLÉANS Cedex 2

TÉL | (33) (0)2 38 41 70 37

MAIL | leo@univ-orleans.fr

www.leo-univ-orleans.fr
# Should I Wait or Should I Pay?
## The Dynamics of Private and Public Healthcare*

Denis Caporale<sup>a</sup>, Francesco Magris<sup>b,c</sup>, Gabriele Sbaiz<sup>b,d</sup>, Bianca Tanasa<sup>b,c,†</sup>

### Abstract

In many countries, a private health system coexists with the public one. Recently, however, it seems that the latter is subjected to budgetary cuts and no longer able to meet the needs of citizens. Within an *OLG* model with production and in which young agents invest in preventive medicine and consume private healthcare beside public one, we prove that access to public health system is rationed by increasing waiting times, entailing costs in terms of treatment effectiveness and psychological order. Considering a fiscal policy aimed at providing health services, we prove that there emerge two dynamic regimes, the first prevailing for low *GDP* levels, the other for larger ones, where waiting times are counter-cyclical. We find two stationary capital levels, the lower one being globally unstable while the higher one globally stable. We also study the set of optimal allocations and prove that stationary *GDP* is increasing in the social discount factor. Finally, we show that it is possible to decentralize any optimal allocation by fixing opportunely the debt-to-*GDP* ratio and government spending in healthcare.

**JEL Classification:** C61; E62; I12; I18

**Keywords:** Aggregate Welfare; Global Dynamics; Fiscal Policy; Public Healthcare Sustainability; Waiting Times

<footnote>*We would like to thank Luciana Bartolini, Paolo Berardinelli, Gianni Bosi, Roberto Caporale, Cristian Ferraris, Raquel Fonseca, Esteban Indurain, Alberto Mingardi, Mihai Mutascu, Daria Onori, Cristina Strango and Simone Tonin for helpful comments and suggestions. We also wish to thank TIMTED 2025 (Timişoara, Romania), AMASES 2025 (Florence, Italy) and IMEWS 2025 (Pamplona, Spain) participants. Any remaining errors are our own.</footnote>

<footnote><sup>a</sup> General Manager of the entity Azienda Sanitaria Universitaria Friuli Centrale (ASUFC), Italy.</footnote>

<footnote><sup>b</sup> Department of Economics, Business, Mathematics and Statistics (DEAMS), University of Trieste, Italy.</footnote>

<footnote><sup>c</sup> Laboratoire d'Économie d'Orléans (LEO), University of Orléans, France.</footnote>

<footnote><sup>d</sup> INdAM – National Institute for Advanced Mathematics, Rome, Italy.</footnote>

<footnote><sup>†</sup> Corresponding author. Via Alfonso Valerio, 4/1 – 34127 Trieste (Italy); Rue de Blois, BP 26739 – 45067 Orléans (France). Email: bianca.tanasa@phdunits.it.</footnote>
# 1 Introduction

In this work, our principal aim is to study the short and long-term dynamics of a public health-
care system coexisting alongside a private one. In particular, we want to emphasize how, on the
one hand, the rationing mechanism of private healthcare is entrusted to the price system while,
on the other hand, the one of public healthcare is entrusted to waiting lists. In this context, we
evaluate the losses in efficiency of a market system and how the former can be restored through
the fiscal policy, in particular the public debt management.

It is widely recognized that the current configuration of European health systems lies at
the intersection of two opposing forces: on the one hand, the universalistic tradition of na-
tional health services, aimed at ensuring equity of access and coverage regardless of individuals'
economic condition; on the other, the growing diffusion of private channels, which represent a
feasible alternative especially when public sector waiting lists become excessively long. The pos-
sibility of turning to the private sector, however, is not uniform, as it depends on the economic
capacity of individuals. This dualism has given rise to hybrid forms in almost all European coun-
tries, in which the public component remains predominant although it is subject to increasing
fiscal constraints, while the private one acts as a selective outlet valve.

## 1.1 Literature Review

The theoretical ground of waiting lists as a rationing device dates back to the fundamental con-
tribution of Arrow (1978) who, while not directly addressing waiting lists issues, introduces the
key concepts explaining why the healthcare market cannot work as a standard competitive one.
Uncertainty, information asymmetry between doctors and patients, as well as incomplete con-
tracts imply that non-market institutions (non-profit hospitals, insurance, regulations) emerge
spontaneously to fill these failures. It is on this ground that the theory of waiting lists does take
relevance.

One of the first specific contribution is given by Barzel (1974), who interprets waiting time
as a time price, namely a shadow price balancing demand with supply when monetary prices
cannot play that role. The queue thus becomes the main device through which the health system
replaces the market price, introducing non-monetary rationing. This is formalized by Lindsay
and Feigenbaum (1984), who model entry into the list as an economic decision: a more efficient
public healthcare system reduces individual waiting times but can lengthen the overall size of
the list, showing the intrinsic inefficiency of the system. Cullis and Jones (1986) criticize the
interpretation of time as a price, pointing out that waiting generates additional social costs, both
in terms of health and individual welfare. In a later work, Cullis et al. (2000) consolidate this
interpretation of waiting lists: they are not simply organizational malfunctions, but allocation
tools with complex redistributive and efficiency effects.

On the ground of these theoretical pillars, the literature has been enriched with increasingly
refined models. Iversen (1993) builds a non-cooperative model of hospital allocation in a na-
tional health system, showing how waiting lists emerge endogenously in view of the interaction of
budget constraints and institutional incentives. The same author extends the analysis to include
the private sector and dual practice, showing that the possibility for doctors to work also in the
private sector can lengthen waiting times in the public sector in the absence of adequate regu-
lation (Iversen, 1997). At the same time, Hoel and Sæther (2003) offer an original perspective:
longer waiting times can entail positive redistributive effects, because they induce self-selection
among individuals with different willingness to pay or different tolerance for waiting. In this
sense, waiting lists become an implicit tool for vertical equity.

An alternative approach is that of Farnworth (2003) who introduce a game-theoretic model
in which health care facilities compete on both prices and waiting times: an increase in prices in
some facilities can reduce overall waiting times, suggesting that co-payments or tariffs may be useful demand management tools. Siciliani (2007) uses a principal-agent model, showing that optimal contracts may involve separating equilibria in which hospitals with high demand end up with longer waiting times. Discrimination in waiting times thus is not necessarily inefficient, but may be a mechanism for allocating resources under informational constraints. Gravelle and Siciliani (2009) further develop this point, proving that third-degree discrimination in waiting times can maximize social welfare. More recent contributions have enriched the mathematical approach of the issue. In particular, Bonneuil (2021) develop an optimal management model of ventilator queues during the Covid crisis, showing that rules based on age and sex can minimize mortality weighted by life expectancy. Xiao et al. (2024), on the other hand, propose a robust optimization approach to manage overbooking in the presence of no-shows and cancellations, emphasizing how robust booking policies can reduce both average waiting times and variability.

On the empirical ground, the literature has sought to test and quantify the above mentioned
theoretical insights. Propper (1995) uses valuation methods to monetize the disutility of time
spent in queues, showing that waiting has measurable welfare costs. Besley et al. (1999) show
that in Britain longer waiting lists are correlated with greater demand for private insurance,
a result replicated by Jofre-Bonet (2000) for Spain and by Pizer and Prentice (2011) for the
U.S. veterans system. More recent evidence, such as that of the one by Yang et al. (2024) for
Australia, confirms that private insurance can alleviate, although to a limited extent, public
waiting times.

At the same time, Martin and Smith (1999) use panel data to show that incentives and
demand pressure are the true determinants of waiting lists, more than mere hospital capacity.
Riganti et al. (2017) replicate this result in Italy, showing demand to be inelastic with respect
to waiting times and supply to be reactive, while Brindley et al. (2023) estimate in England
a causal effect of spending on waiting times, even if limited in magnitude: a 1% increase in
spending reduces average waiting times by only 0.6 days. Sivey (2018) shows that in Australian
emergency departments long queues reduce demand for non-urgent treatments, confirming the
implicit rationing function of waiting. Eventually, McQuestin and Noguchi (2020) analyze the
impact of government funding on emergency waiting times, finding positive but short-lived
effects.

The policy dimension of waiting lists has been systematically addressed by Siciliani et al. (2013), who in a large comparative study show that waiting time reduction policies work only if multilayered: isolated targets risk generate opportunistic behavior (gaming), while integrated packages including clinical prioritization, output incentives, transparent monitoring and purchase of additional capacity are more effective. Siciliani et al. (2015) update and deepen these results, reiterating that sustainability requires robust institutions and stable reporting. Other contributions discuss the role of the private sector in alleviating or aggravating public waiting times. For example, Duckett (2005) analyzes the Australian case, where the private sector can relieve waiting lists but also crowd out public resources, while Mou (2013) show how inequality and aging politically reduce the space for public spending, favoring the growth of the private sector.

Another important strand of literature studies the effects of waiting lists on equity and macro-sustainability. Costa-Font and Cowell (2013) show the importance of using inequality measures consistent with the ordinal nature of health data: waiting times hit hardest those with poorer resources, and inadequate metrics risk underestimating this effect. Doetsch et al. (2023) and Quaglio et al. (2013) show how austerity policies have increased access problems, especially for vulnerable groups. Varbanova et al. (2023) underline that reducing waiting times is only one lever affecting macro-outcomes such as life expectancy and DALYs (Disability-Adjusted Life Years). Kapteyn (2010) warns against superficial international comparisons, stressing how
different institutions produce different effects of waiting.

Waiting lists, particularly for outpatient services and elective treatments, are not a transitory phenomenon, but rather a structural feature of public health systems, as widely documented by empirical and comparative studies (Martin and Smith, 1999; Siciliani et al., 2013, 2015). They not only reflect the organizational and financial limits of public healthcare, but also, at the macroeconomic level, raise a problem in terms of overall sustainability: excessive pressure on the public sector takes the risk of generating widespread access barriers, with aggregate consequences on the system's ability to ensure timely and equitable care. In this context, individual decisions to resort to the private sector become an adjustment mechanism that, although reliable, does not eliminate the underlying critical issues, but rather tends to shift part of the demand outside the public system (Besley et al., 1999; Jofre-Bonet, 2000; Pizer and Prentice, 2011; Yang et al., 2024).

## 1.2 Paper Contribution

To shed additional light on the dynamics of waiting lists and on the sustainability of public healthcare systems, in this work we study a two-period OLG (Overlapping Generations) model in which young people invest their labor income, besides productive capital and government bonds, also in healthcare. More specifically, health is an individual specific good evaluated only by old people; however, young agents can undertake investment in preventive medicine allowing them to reach retirement age within a more healthy status. The accumulated health status will in turn determine the efficiency of public and private healthcare services used during retirement age and thus their overall impact on individual welfare.

Once a young agent has made her investment choices, when old she will face two types of trade-offs. On the one hand, she will have to decide how to allocate her savings between “ordinary” consumption and healthcare spending; on the other, she will have to decide where to satisfy her demand of healthcare between the private and public sectors. If the cost of private healthcare is measured by its market price, public healthcare is instead provided free of charge by the government. In the absence of any private cost to bear, the demand for public healthcare would therefore be driven up to infinite. Thus, we introduce a rationing mechanism based on longer waiting times induced by large demand for public healthcare. Waiting times represent indeed a dual cost: one of psychological order due to the time lost and one of physical order as the decreasing effectiveness of a given medical treatment deferred over time. The performance of any medical treatment, in fact, is the composite outcome of the skills of the medical personnel, the efficiency of the technologies used, and of the administration's speed, as every pathology worsens over time.

Within the market economy, we find that equilibrium dynamics, for low levels of income, entails no demand for private healthcare since public healthcare provision results to be large enough compared with agents income: they prefer to allocate their scarce resources to consumption. It is only when income increases enough that agents begin to consume private healthcare alongside public one. The equilibrium dynamics is thus represented by the superior envelope of two different dynamics. We find two stationary solutions ranked with respect to the capitalist intensity. The lower stationary solution is proven to be globally unstable, the higher one being globally stable. The main economic mechanisms at work clearly emerge when we simulate equilibrium dynamics on the ground of empirically consistent calibrations. Actually, standard real variables are procyclical, while waiting times are countercyclical since wealthier agents decide to switch to the private healthcare sector when faced with excessively long waiting times, with the result of easing the pressure of the aggregate demand on public healthcare system.

It is well known that welfare analysis in OLG models requires the introduction of aggregate welfare functions defined on the utilities of all generations. To this end, we study the allocations
chosen by a social planner discounting future. This allows identifying the set of Pareto-efficient allocations and prove that they are characterized by a capitalist intensity increasing in social patience.

Finally, we show how to manage the fiscal tools appropriately to obtain any Pareto-optimal allocation as the outcome of the market economy. This in particular requires, under mild restrictions on the structural parameters, to adequately calibrate the debt-to-capital ratio. However, and rather surprisingly, we do not obtain a monotonic relationship between the debt-to-capital ratio required to decentralize Pareto-optima and degree of social patience. We get instead a bell-shaped curve such that for low income levels, an increase in the targeted capital requires an increase in the debt-to-capital ratio while above a certain income threshold, a further increase in the targeted capital requires instead a decrease in the debt-to-capital ratio. Such a configuration is easily understandable once one observes that low capital targets in the market economy are implemented through low-stationary states, whose capital intensity is increasing in the debt-to-capital ratio. By contrast, larger capital stocks are implemented through high-stationary states, which are instead decreasing in the debt-to-capital ratio.

To compare our work with the literature above mentioned, note that the latter shares a common thread: waiting lists are a structural mechanism, with solid theoretical roots and wide empirical confirmation, but their effects and possible solutions depend heavily on institutional and fiscal context. In particular, if earlier models interpreted waiting as a shadow price (Barzel, 1974; Lindsay and Feigenbaum, 1984), later contributions clarified that they should viewed in relation to incentives, budget constraints, and the role of the private sector (Cullis and Jones, 1986; Iversen, 1993, 1997; Hoel and Sæther, 2003). Empirical evidence further confirmed that recourse to the private sector does not eliminate the problem, but rather tends to shift demand outside the public system selectively, without solving its structural knots (Besley et al., 1999; Jofre-Bonet, 2000; Propper, 1995; Yang et al., 2024).

Our work fits within this literature tradition, although introduces an additional level of analysis linked to the intertemporal and fiscal dimension. In particular, in our *OLG* theoretical model, individuals decide whether to resort to the public sector, facing potentially longer waiting times, or to the private sector, paying higher monetary costs. The decentralized structure of the model embeds two distinct regimes: a binding one, in which public health care effectively constrains individual choices through waiting lists, and a non-binding regime, in which private health care becomes the dominant option. This distinction recalls and generalizes the logic of rationing-by-queue models, but integrates it with the possibility of switching between regimes depending on fiscal parameters and macroeconomic conditions.

One of the main novelties of our contribution lies in the introduction of public debt as a rebalancing instrument. While the existing literature has often analyzed waiting lists in a static context or with exogenous budget constraints, here debt becomes an endogenous policy tool, through which the government can modulate the tax burden on the young and the level of collectively financed health spending. In particular, debt takes on an intertemporal stabilization function: for low capital values, an increase in public debt makes it possible to support health spending and keep the public system accessible; for high capital values, by contrast, excessive debt can generate explosive dynamics, increasing future tax pressure and reducing the overall sustainability of the system. This dynamics thus introduces a new tension compared to existing models: not only waiting becomes a shadow price regulating access, but debt itself acts as a mechanism for shifting health costs across generations, influencing both the individual choices between public and private and the long-term macroeconomic stability.

Our model therefore does not simply describe the coexistence of public and private as in Brekke and Sørgard (2007) or Hoel and Sæther (2003), but proposes an integrated framework in which fiscal policy, individual choices, and waiting lists are analyzed jointly. The methodological
novelty lies in the application of an *OLG* model with multiple regimes, capable of explaining how critical thresholds of debt and capital can determine transitions between sustainable public health systems and collapse scenarios, in which public health spending explodes and the system becomes inaccessible even at an aggregate level.

In this sense, our contribution is twofold: on the one hand, it provides a theoretical generalization of classical rationing models, incorporating fiscal constraints and debt; on the other, it offers an interpretive framework linking the microeconomics of individual health choices with the macro-fiscal sustainability of mixed systems, filling a gap in the existing literature. The work thus draws on Arrow (1978) insights on uncertainty, Barzel (1974) and Lindsay and Feigenbaum (1984) on rationing through waiting, and subsequent extensions on public-private interaction (Iversen, 1997), to propose a new theoretical perspective in which public debt becomes a key variable for understanding the equilibrium conditions and intertemporal sustainability of health care.

## 1.3 The Case of Friuli-Venezia Giulia

On the one hand, the theoretical literature has clarified the reasons why waiting lists emerge as a rationing tool in public health systems (Barzel, 1974; Lindsay and Feigenbaum, 1984; Cullis and Jones, 1986); on the other, a wide body of empirical work has evaluated their concrete impact both on individual behavior and on the overall efficiency of health systems. Studies conducted in different national contexts show how longer waiting times are associated with a greater probability of purchasing private insurance (Besley et al., 1999; Jofre-Bonet, 2000; Pizer and Prentice, 2011; Yang et al., 2024) and how such individual choices translate into redistributive effects that, however, do not eliminate the structural weaknesses of the public sector. At the same time, works such as Propper (1995); Street and Duckett (1996) have quantified the disutility of time spent in queues, while more recent analyses on the Italian context have shown that waiting lists remain a crucial dimension in the evaluation of health performance (France et al., 2005; Riganti et al., 2017).

The Italian case represents a particularly fertile ground for observing these dynamics. The National Health Service, although universalistic, is characterized by strong regional heterogeneity in the organization and delivery of care. This heterogeneity is reflected above all in waiting lists, where alongside success stories there are still wide criticalities. The most recent empirical data confirm what has been observed at a comparative level: while targeted interventions have produced significant improvements in some high-priority clinical areas, chronic difficulties persist in ensuring acceptable access times for outpatient services.

The analysis performed on the case of Friuli Venezia Giulia (ASUFC) clearly highlights this duality. In recent years, the regional system has achieved significant progress in managing waiting lists for oncological pathologies considered urgent, reaching standards increasingly close to those set by national guidelines. These improvements testify the ability of the public system to concentrate resources and efforts where the clinical and social impact of waiting is greatest.

However, the same process is not observed as effectively in outpatient services, where waiting times remain systematically high and constitute one of the main sources of pressure for both patients and policymakers. This phenomenon reflects what has also emerged from the international literature: increasing capacity or introducing sectoral targets reduces waiting in some areas, but does not solve the problem structurally (Martin and Smith, 1999; Siciliani et al., 2015; Brindley et al., 2023). In the absence of a systemic rethinking of financing and resource allocation mechanisms, improvements remain partial and do not prevent the formation of new pockets of inefficiency.

This empirical evidence reinforces the need for a theoretical approach that can hold together the different dimensions of the problem: on the one hand, clinical urgency, which justifies
immediate and targeted interventions; on the other, macro-fiscal sustainability, without which no local improvement can be maintained over time. In this sense, the model developed in the present work, by integrating the choice between public and private health care with the intertemporal constraint of debt, is able to offer an analytical framework fitted to interpreting real data and suggesting equilibrium conditions able to guarantee not only efficiency but also long-term sustainability of a mixed health system.

Figures 1 and 2 show the evolution of waiting lists in the Friuli Venezia Giulia Region, with particular reference to the University Health Authority of Friuli Centrale (ASUFC). In Figure 1, the charts relating to oncology treatments show a significant improvement in compliance with the maximum times set at national level: in almost all departments, the percentage of treatments carried out within the time limit is above 90%. This indicates the ability of the regional health system to concentrate resources and efficiency in areas of highest clinical priority. A different picture emerges for outpatient and diagnostic services, where average waiting times remain high and the share of appointments carried out within the limits remains unsatisfactory, as shown in Figure 2. These data confirm the structural criticalities already highlighted by the international literature, namely the difficulty of uniformly reducing waiting lists by acting only on specific sectors.

In summary, the experience of Friuli Venezia Giulia suggests that targeted interventions can produce tangible improvements in urgent areas such as oncology, but are not sufficient to ensure overall sustainability. Inefficiencies persist in outpatient services, which represent the main source of pressure for both citizens and policymakers. Such empirical evidence motivates the theoretical analysis developed in this work, in which the interaction between public and private, fiscal constraints and debt dynamics are modeled in order to understand the equilibrium conditions and long-term sustainability of the health system.

<figure><img src="image_5.png" /><figcaption>Figure 1: Oncology Treatments</figcaption></figure>
<table><thead><tr><th colspan="9">MONITORING</th></tr><tr><th colspan="4">January - June 2024</th><th colspan="4">January - June 2025</th></tr><tr><th>Diagnostic Imaging</th><th>Appointments to be guaranteed</th><th>Target Appointments</th><th>% within maximum waiting time</th><th>Appointments to be guaranteed</th><th>Target Appointments</th><th>% within maximum waiting time</th><th>Trend in target</th><th>Delta Appointments</th><th>Delta Appointments</th></tr></thead><tbody><tr><td>CT scan</td><td>8267</td><td>5856</td><td>70,8%</td><td>7461</td><td>5224</td><td>70,0%</td><td>▼</td><td>▼</td><td>-10,8%</td></tr><tr><td>MRI</td><td>8949</td><td>5558</td><td>62,1%</td><td>9383</td><td>7342</td><td>78,2%</td><td>▲</td><td>▲</td><td>4,6%</td></tr><tr><td>Ultrasound Scan</td><td>28462</td><td>20339</td><td>71,5%</td><td>27736</td><td>21629</td><td>78,0%</td><td>▲</td><td>▼</td><td>-2,6%</td></tr><tr><td>Mammography + Ultrasound</td><td>7042</td><td>3665</td><td>52,0%</td><td>6269</td><td>4702</td><td>75,0%</td><td>▲</td><td>▼</td><td>-12,3%</td></tr><tr><td>TOTAL</td><td>52720</td><td>35418</td><td>67,2%</td><td>50849</td><td>38897</td><td>76,5%</td><td>▲</td><td>▼</td><td>-3,7%</td></tr></tbody></table>

<table><caption>Figure 2: Outpatient Services</caption><thead><tr><th colspan="9">MONITORING</th></tr><tr><th colspan="4">January - June 2024</th><th colspan="4">January - June 2025</th></tr><tr><th>Other Specialist Examinations</th><th>Appointments to be guaranteed</th><th>Target Appointments</th><th>% within maximum waiting time</th><th>Appointments to be guaranteed</th><th>Target Appointments</th><th>% within maximum waiting time</th><th>Trend in target</th><th>Delta Appointments</th><th>Delta Appointments</th></tr></thead><tbody><tr><td>Colonoscopy</td><td>1650</td><td>374</td><td>22,7%</td><td>1739</td><td>382</td><td>22,0%</td><td>▼</td><td>▲</td><td>5,1%</td></tr><tr><td>Gastroscopy</td><td>2383</td><td>753</td><td>31,6%</td><td>2285</td><td>623</td><td>27,3%</td><td>▼</td><td>▼</td><td>-4,3%</td></tr><tr><td>Electrocardiogram</td><td>4797</td><td>4302</td><td>89,7%</td><td>4880</td><td>4639</td><td>95,1%</td><td>▲</td><td>▲</td><td>1,7%</td></tr><tr><td>Dynamic Electrocardiogram (Holter)</td><td>938</td><td>608</td><td>64,8%</td><td>924</td><td>431</td><td>46,6%</td><td>▼</td><td>▼</td><td>-1,5%</td></tr><tr><td>Cardiac Stress Test</td><td>643</td><td>544</td><td>84,6%</td><td>477</td><td>265</td><td>55,6%</td><td>▼</td><td>▼</td><td>-34,8%</td></tr><tr><td>Pure Tone Audiometry</td><td>2000</td><td>1147</td><td>57,4%</td><td>2162</td><td>1317</td><td>60,9%</td><td>▲</td><td>▲</td><td>7,5%</td></tr><tr><td>Spirometry</td><td>1562</td><td>784</td><td>50,2%</td><td>1449</td><td>975</td><td>67,3%</td><td>▲</td><td>▼</td><td>-7,8%</td></tr><tr><td>Fundus Photography</td><td>49</td><td>19</td><td>38,8%</td><td>138</td><td>0</td><td>0,0%</td><td>▼</td><td>▲</td><td>64,5%</td></tr><tr><td>Electromyography</td><td>2404</td><td>745</td><td>31,0%</td><td>211</td><td>110</td><td>52,1%</td><td>▲</td><td>▼</td><td>-1039,3%</td></tr><tr><td>TOTAL</td><td>16426</td><td>9276</td><td>56,5%</td><td>14265</td><td>8742</td><td>61,3%</td><td>▲</td><td>▼</td><td>-15,1%</td></tr></tbody></table>

The remainder of the paper is organized as follows. In Section 2 we study the market economy while Section 3 is devoted to the characterization of Pareto-optimal allocations. In Section 4 we perform the decentralization of the Pareto-optima in the market economy and Section 5 concludes the paper.

## 2 The Market Economy

We consider an *OLG* economy where in each period $t = 0, 1, 2, \dots$ coexist young agents beside old ones, a representative firm, and a public sector. The size of each generation of households is normalized to one and there is no population growth. Let us now describe in detail the behavior of each economic unit within the decentralized economy, where firms behave competitively and no altruistic rationale is accounted.
## 2.1 Government

At each date $t$, government has to pay the gross interest rate $R_t$ on the real bonds $B_t^G$ previously issued and per-capita public spending $g_t$. To finance these expenditures, it levies per capita taxes $\tau_t$ and issues new bonds so that in each date $t$ its budget constraint in per-capita terms is respected

$$B_{t+1}^G = R_t B_t^G + g_t - \tau_t.$$

We assume that all public spending $g_t$ is employed to finance public health $n_t^{ph}$. Public spending on healthcare is considered to be both divisible and rival. Furthermore, we assume the efficient units of public healthcare spending to be larger than those of private healthcare one. In particular, if we take one unit of public healthcare $n_t^{ph}$, one has $\delta > 0$ units of public spending $g_t$, i.e.

$$\delta n_t^{ph} = g_t. \qquad (1)$$

Thus, the parameter $\delta$ represents the (constant) marginal rate of transformation between public and private healthcare. We introduce a restriction on $\delta$ to ensure a positive solution for public healthcare demand, which would otherwise be driven to zero and even become negative in view of the quadratic cost it entails. It is easy to prove that this requires public healthcare to be more efficient than private one, i.e. $\delta < 1$. Therefore we have

**Assumption 1.** $0 < \delta < 1$.

Let us assume that government sets the amount of constant public spending $g_t = \bar{g}$ at each date $t$ so that the amount $\bar{n} = \delta^{-1}\bar{g}$ of public healthcare is constant too. In addition, government aims at stabilizing the public debt-to-capital ratio, i.e. it sets $B_t^G/k_t = \hat{b}$, so that

$$B_t^G = \hat{b}k_t. \qquad (2)$$

Note that the debt-to-capital ratio can be interpreted as a proxy for the public debt-to-GDP<fnref n="1" /> ratio, the level of which is easily found in the official statistics of many countries. Furthermore, there are also thresholds imposed on this ratio, such as the 60% threshold in Europe following the Maastricht Agreements. Therefore, we can write the dynamics of public debt as

$$\hat{b}k_{t+1} = R_t \hat{b}k_t + \delta \bar{n} - \tau_t,$$

to obtain the feedback rule in terms of lump-sum taxes:

$$\tau_t = \hat{b}(R_t k_t - k_{t+1}) + \delta \bar{n}. \qquad (3)$$

As it will be clear in the sequel, the two fiscal parameters represented by public spending $\bar{g}$ and the public debt stabilization rule $\hat{b}$, will allow us on the one hand to carry out comparative statics exercises and on the other one to decentralize the set of optimal solutions obtained by maximizing social welfare.

## 2.2 Firms

We assume that a representative competitive firm operates in the economy with an infinite time horizon. In each period $t$, the firm employs capital $k_t$ and labor $l_t$ to produce a homogeneous good that can be consumed $c_t$, invested in productive capital $k_{t+1}$, in preventive medicine $I_t$, or

<footnote n="1">¹GDP stands for Gross Domestic Product.</footnote>
used for healthcare services, either private $l_t$ or public $n_t$. Technology is of the Cobb-Douglas type with constant returns to scale such that GDP is given by

$$y_t = k_t^\gamma l_t^{1-\gamma} \quad (4)$$

with $\gamma \in (0, 1)$ representing the share of capital in total income. As a consequence, static profit maximization conditions yield the following expressions for the gross real interest rate $R_t$ and the real wage $w_t$:

$$R_t - (1 - \eta) = \gamma (k_t/l_t)^{\gamma-1} \quad (5)$$

and

$$w_t = (1 - \gamma) (k_t/l_t)^{\gamma} \quad (6)$$

where $\eta \in [0, 1]$ is the constant capital depreciation rate. We now describe households' behavior.

## 2.3 Households

In each period $t$ there is a continuum of representative young agents as well as old ones, whose sizes are unitary. We can thus denote each individual belonging to a given generation with the index $i \in [0, 1]$. For the purpose of our analysis and in order to keep the model analytically tractable without losing in generality, we assume the utility function of a young agent $i$ born in period $t$ to be defined only on his old age consumption. More specifically, an old agent $i$ born in $t$ derives utility from her consumption $c_{t+1}(i)$ of ordinary good and from her health status $A_{t+1}(i)$ ($l_{t+1}(i) + n_{t+1}(i)$), which is the product of her health accumulation $A_{t+1}(i)$ (consequence, as we will explain soon, of the investment $I_t(i)$ in preventive medicine undertaken when young) with the total amount of and her private $l_{t+1}(i)$ as well as public $n_{t+1}(i)$ healthcare spending when old.

As we will see later on, private health care spending affects the budget constraint of old agents, while public health care is provided free of charge by the government. Therefore, to avoid an infinite demand for public health care and introduce a rationing mechanism, we make the hypothesis that public healthcare consumption entails a disutility $T_{t+1}(i)$. Such a disutility is assumed to be increasing in aggregate demand $n_{t+1}$ of public healthcare, although in a time-dependent way. As a matter of fact, the complete utility function of an agent $i$ born in period $t$ is given by

$$U(i) = \ln c_{t+1}(i) + \alpha \ln[A_{t+1}(i)(l_{t+1}(i) + n_{t+1}(i))] - T_{t+1}(i) \quad (7)$$

where $\alpha > 0$ denotes the agents' propensity for enjoying a good health status.

We assume that $T_{t+1}(i)$ grows at increasing rates with average public health demand $(\int_0^1 n_{t+1}(i)f(i)di) / (\int_0^1 f(i)di)$, where $n_{t+1}(i)$ is the demand of public healthcare of individual $i$ and $f(i)$ denotes households' density function. $T_{t+1}(i)$ grows with average public health demand in view of the longer waiting times generated by larger per-capita levels of public health demand. Longer waiting times indeed lead to psychological distress and, above all, a deterioration in the effectiveness of the health care service itself, as postponing treatment leads to a worsening of patients' health status. We assume $T_{t+1}(i)$ to be quadratic in per-capita demand of public healthcare:

$$T_{t+1}(i) = \beta_{t+1} \frac{1}{2} \left[ \frac{\int_{0}^{1} n_{t+1}(i) f(i) di}{\int_{0}^{1} f(i) di} \right]^2,$$

where the time-dependent component $\beta_{t+1} > 0$ of waiting times disutility is discussed later. Since agents are identical and defined on the unit interval, it is immediate to verify that
$\int_0^1 f(i)di = 1$ and $\int_0^1 n_{t+1}(i)f(i)di = n_{t+1}$, where $n_{t+1}$ is the common individual demand for public health. It follows that for every $i \in [0, 1]$ one has

$$T_{t+1}(i) = \beta_{t+1} \frac{1}{2} n_{t+1}^2. \qquad (8)$$

Each individual is thus aware of the impact of her demand of public healthcare on the lengthening of waiting times, given the demand of the other agents. However, at symmetric equilibrium all choices are identical, even if each individual, at the moment of maximizing, ignores it. From now on, when describing the individual behavior, we can drop the index $i$.

Finally, the term $\beta_{t+1}$ impacting the disutility of waiting times (that agents take as given) is time dependent. As we will see, this term adjusts endogenously over time to align the demand for public health with its supply and represents a rather innovative mechanism of our work. In summary, we can write the utility function of the representative agent born at time $t$ as

$$U(c_{t+1}, A_{t+1}, l_{t+1}, n_{t+1}) = \ln c_{t+1} + \alpha \ln[A_{t+1}(l_{t+1} + n_{t+1})] - \beta_{t+1} \frac{1}{2} n_{t+1}^2. \qquad (9)$$

A young agent born in $t$ supplies inelastically one unit of labor and must decide how much to invest in capital $k_{t+1}$, in public real bonds $B_{t+1}$ and in preventive medicine $I_t$ out of her net income $w_t - \tau_t$, where $w_t$ stands for the unitary real wage and $\tau_t$ for lump sum taxes. Her young age budget constraint is thus

$$k_{t+1} + B_{t+1} + I_t = w_t - \tau_t. \qquad (10)$$

When old, an agent born in $t$ finances her expenditures in the consumption good as well as in private health out of capital and bonds income, i.e.

$$c_{t+1} + l_{t+1} = R_{t+1} (k_{t+1} + B_{t+1}), \qquad (11)$$

where $R_{t+1}$ stands for the gross interest rate. Note that the return on equity holding and the return on the financial asset must be equal under usual arbitrage conditions. Finally, old age health status $A_{t+1}$ depends on young age investment $I_t$ in preventive medicine according to

$$A_{t+1} = \phi I_t, \qquad (12)$$

where the parameter $\phi > 0$ represents the productivity of preventive medicine, i.e. its capability to convert into health status. In light of equation (12), we can interpret the investment in preventive medicine as all those expenses by young people aimed at taking care of their health, following a healthy lifestyle like eating well, exercising, undergoing periodic medical check-ups. All these efforts will be rewarded in old age by a larger health status $A_{t+1}$.

Since, in light of the equation (9), the utility of the overall health status of the elderly is given by the product of the basic health status $A_{t+1}$ with the sum $l_{t+1} + n_{t+1}$ of private and public spending on healthcare, there could in principle be negative solutions either for $l_{t+1}$ or $n_{t+1}$. Being the supply of public healthcare exogenously fixed at a positive level, a non-negativity constraint must be imposed on private healthcare demand, i.e.

$$l_{t+1} \ge 0. \qquad (13)$$

The problem of an individual born at time $t$ thus consists in choosing $(k_{t+1}, I_t, c_{t+1}, A_{t+1}, l_{t+1}, n_{t+1})$ to maximize (9) under constraints (10), (11), (12) and (13). By assuming constraint (13) to do not bind and replacing $I_t = A_{t+1}/\phi$ in (9) and (10), after some straightforward
computations allowing to get rid of the Lagrangian multipliers, we obtain the following First Order Conditions (FOCs)

$$l_{t+1} + n_{t+1} = \alpha c_{t+1}, \qquad (14)$$

$$\phi \alpha c_{t+1} = A_{t+1} R_{t+1}, \qquad (15)$$

and

$$\alpha = \beta_{t+1}(l_{t+1} + n_{t+1})n_{t+1}, \qquad (16)$$

with constraints (10), (11) and (12). By contrast, when constraint (13) binds, one must set $l_{t+1} = 0$ and then reformulate the whole optimization problem in terms of the remaining variables. In such a case, beside constraints (10), (11) and (12), one obtains the alternative FOC's which, respectively, link consumption and waiting times to public health expenditures:

$$c_{t+1} = n_{t+1}/\alpha, \qquad (17)$$

and

$$\beta_{t+1} = n_{t+1}^2/\alpha. \qquad (18)$$

We can now couple government, firms and households behavior with the market clearing conditions to obtain the dynamic system describing intertemporal equilibrium of the market economy.

## 2.4 Intertemporal Equilibrium

In each period $t$ there are four markets in equilibrium: the goods one, the factors (capital and labor) ones and the securities one. To ensure that, one must set $B_t = B_t^G$ and $l_t = 1$ in (2), (3), (4), (5), (6), (10), (11), (14), (15) and (16). Assuming full capital depreciation, i.e. $\eta = 1$, when constraint (13) is not binding, coupling the corresponding FOCs with the market clearing ones, one has $c_{t+1}$, $l_{t+1}$, $\beta_{t+1}$ and $A_{t+1}$ (and thus $I_t$) as functions only of $k_{t+1}$. Indeed

$$c_{t+1} = \frac{\gamma k_{t+1}^{\gamma} (1 + \hat{b}) + \bar{n}}{1 + \alpha}, \qquad (19)$$

$$l_{t+1} = \alpha \frac{\gamma k_{t+1}^{\gamma} (1 + \hat{b}) + \bar{n}}{1 + \alpha} - \bar{n}, \qquad (20)$$

$$\beta_{t+1} = \frac{1 + \alpha}{\bar{n}[\gamma k_{t+1}^{\gamma} (1 + \hat{b}) + \bar{n}]}, \qquad (21)$$

and

$$A_{t+1} = \phi I_t = \frac{\alpha \phi [\gamma k_{t+1}^{\gamma} (1 + \hat{b}) + \bar{n}]}{\gamma k_{t+1}^{\gamma-1} (1 + \alpha)}. \qquad (22)$$

From (20) it is immediate to verify that $l_{t+1} > 0$, and thus that constraint (13) does not bind, if and only if the capital stock is large enough, namely if and only if

$$k_{t+1} > k^{TH} \qquad (23)$$

where

$$k^{TH} = \left( \frac{\bar{n}}{\alpha \gamma (1 + \hat{b})} \right)^{\frac{1}{\gamma}} \qquad (24)$$

denotes the critical threshold of capital separating the binding regime from the non-binding one. As a consequence, under condition (23), equations (19), (20), (21) and (22) are consistent with
$l_{t+1} > 0$. Combining them with the budget constraint (10) and the fiscal rules, we obtain the following first-order difference equation describing the dynamics of capital accumulation:

$$ \left( 1 + \frac{\alpha(1 + \hat{b})}{(1 + \alpha)} \right) k_{t+1} + \frac{\alpha \bar{n}}{\gamma(1 + \alpha)} k_{t+1}^{1-\gamma} + \bar{n} \bar{\delta} = \left( 1 - \gamma(1 + \hat{b}) \right) k_t^{\gamma}. \quad (25) $$

By contrast, when $k_{t+1} \le k^{TH}$, constraint (13) binds and the corresponding households' FOC's are (17) and (18). Combining them with the market clearing conditions and the fiscal rules, we get the alternative dynamic system describing intertemporal equilibrium:

$$ (1 + \alpha(1 + \hat{b})) k_{t+1} + \bar{n}\bar{\delta} = (1 - \gamma(1 + \hat{b})) k_t^{\gamma}. \quad (26) $$

Global dynamics describing intertemporal equilibrium of the market economy is thus described by equation (26) for $k_t \in [0, k^{TH}]$ and by equation (25) for $k_t > k^{TH}$. The change in dynamic regime occurs when a capital stock satisfying $k_t < k^{TH}$ is mapped according to (26) to some $k_{t+1} > k^{TH}$. Actually, $k_{t+1}$ must be rather mapped according to (25). Since both (25) and (26) are positively defined in $k_{t+1} \in [0, +\infty)$ when $\hat{b} > -1$ and $1 - \gamma(1 + \hat{b}) > 0$, one must put a lower and upper bound on the debt-to-capital ratio $\hat{b}$ fixed by government. Therefore, one has

**Assumption 2.** $-1 < \hat{b} < (1 - \gamma)/\gamma$.

To get a complete picture of the intertemporal equilibrium encompassing both the binding and non-binding regimes for constraint (13), one thus needs to study equations (25) and (26) separately. In particular, one must find their respective stationary solutions, rank them and, finally, analyze their stability.

### 2.4.1 Equilibrium Dynamics: Non-binding Regime

Let us first analyze the dynamics of the economy under the assumption that constraint (13) does not bind. Checking equation (25), it is easy to invert it to express $k_t$ as a function of $k_{t+1}$:

$$ k_t = F^{NB}(k_{t+1}) \equiv \left( \frac{\left( 1 + \frac{\alpha(1+\hat{b})}{1+\alpha} \right) k_{t+1} + \frac{\bar{n}}{\gamma} \frac{\alpha}{1+\alpha} k_{t+1}^{1-\gamma} + \bar{n}\bar{\delta}}{1 - \gamma(1+\hat{b})} \right)^{\frac{1}{\gamma}}. \quad (27) $$

Equation (27) describes the backward dynamics of capital accumulation under the hypothesis of a non-binding constraint (13). Note that the stationary solutions to (27) are the same as those given by (25). Nevertheless, if a stationary solution is stable (unstable) in the backward dynamics (27), it will be unstable (stable) in the forward one (25). It is worth stressing that, in view of the form of (27), it is possible to perform a global analysis; in particular, the stability properties of the stationary solutions will hold not only locally but globally too.

The next Proposition includes an important preliminary result for in turn characterizing the existence, multiplicity and the global stability of the stationary solutions of (27). In particular, it claims that the function $F^{NB}(k_{t+1})$ defined in (27) is monotonically increasing; it is first concave and then convex after going through an inflection point $\tilde{k}_{t+1}$.

**Proposition 1.** Under Assumption 2, the function (27) is positively defined, continuous and monotonically increasing in $k_{t+1} \in [0, +\infty)$. It is $(\frac{\bar{n}\bar{\delta}}{1-\gamma(1+\hat{b})})^{\frac{1}{\gamma}} > 0$ at $k_{t+1} = 0$ and diverges to infinite when $k_{t+1} \to +\infty$. In addition, there exists an inflection point $\tilde{k}_{t+1} > 0$ such that
$$ \left( \frac{d^2 F^{NB}(k_{t+1})}{dk_{t+1}^2} \right)_{k_{t+1}=\tilde{k}_{t+1}} = 0 \quad \text{and} \quad (27) \text{ is concave in } k_{t+1} \in [0, \tilde{k}_{t+1}) \text{ and convex in } k_{t+1} \in (\tilde{k}_{t+1}, +\infty). $$

*Proof.* It is immediately verifiable that (27) is positively defined and continuous in $k_{t+1} \in [0, +\infty)$ as well as to check its limit points. To proof that it is monotonically increasing, let us compute its first derivative:

$$ \frac{\partial F^{NB}(k_{t+1})}{\partial k_{t+1}} = \frac{1}{\gamma} \left( \frac{\left(1 + \frac{\alpha(1+\hat{b})}{1+\alpha}\right) k_{t+1} + \frac{\bar{n}}{\gamma} \frac{\alpha}{1+\alpha} k_{t+1}^{1-\gamma} + \bar{n}\bar{\delta}}{1 - \gamma(1+\hat{b})} \right)^{\frac{1}{\gamma}-1} \frac{\left(1 + \frac{\alpha(1+\hat{b})}{1+\alpha}\right) + (1-\gamma) \frac{\bar{n}}{\gamma} \frac{\alpha}{1+\alpha} k_{t+1}^{-\gamma}}{1 - \gamma(1+\hat{b})}. \quad (28) $$

It is trivial to verify that (28) is positively valued for all $k_{t+1} > 0$ with $\lim_{k_{t+1} \to 0} \frac{\partial F^{NB}(k_{t+1})}{\partial k_{t+1}} = \lim_{k_{t+1} \to +\infty} \frac{\partial F^{NB}(k_{t+1})}{\partial k_{t+1}} = +\infty$. To prove the existence of an unique inflection point, we write the expression of the second derivative of (27) and set it equal to zero. Straightforward manipulations yield

$$ \begin{aligned} \frac{\partial^2 F^{NB}(k_{t+1})}{\partial k_{t+1}^2} &= \frac{1}{\gamma} \left[ \left( \frac{1}{\gamma} - 1 \right) \left( \frac{\left(1 + \frac{\alpha(1+\hat{b})}{1+\alpha}\right) k_{t+1} + \frac{\bar{n}}{\gamma} \frac{\alpha}{1+\alpha} k_{t+1}^{1-\gamma} + \bar{n}\bar{\delta}}{1 - \gamma(1+\hat{b})} \right)^{\frac{1}{\gamma}-2} \right] \cdot \left[ \frac{1 + \frac{\alpha(1+\hat{b})}{1+\alpha} + (1-\gamma) \frac{\bar{n}}{\gamma} \frac{\alpha}{1+\alpha} k_{t+1}^{-\gamma}}{1 - \gamma(1+\hat{b})} \right] \\ &\quad - \frac{1}{\gamma} \left( \frac{\left(1 + \frac{\alpha(1+\hat{b})}{1+\alpha}\right) k_{t+1} + \frac{\bar{n}}{\gamma} \frac{\alpha}{1+\alpha} k_{t+1}^{1-\gamma} + \bar{n}\bar{\delta}}{1 - \gamma(1+\hat{b})} \right)^{\frac{1}{\gamma}-1} \cdot \frac{\frac{\bar{n}\alpha}{1+\alpha} (1-\gamma) k_{t+1}^{-\gamma-1}}{1 - \gamma(1+\hat{b})} \\ &= 0, \end{aligned} $$

or, equivalently,

$$ \begin{aligned} & \frac{1}{\gamma} \left[ \left( \frac{1}{\gamma} - 1 \right) \left( \frac{\left(1 + \frac{\alpha(1+\hat{b})}{1+\alpha}\right) k_{t+1} + \frac{\bar{n}}{\gamma} \frac{\alpha}{1+\alpha} k_{t+1}^{1-\gamma} + \bar{n}\bar{\delta}}{1 - \gamma(1+\hat{b})} \right)^{\frac{1}{\gamma}-2} \right] \left[ \frac{1 + \frac{\alpha(1+\hat{b})}{1+\alpha} + (1-\gamma) \frac{\bar{n}}{\gamma} \frac{\alpha}{1+\alpha} k_{t+1}^{-\gamma}}{1 - \gamma(1+\hat{b})} \right] \\ & = \frac{1}{\gamma} \left( \frac{\left(1 + \frac{\alpha(1+\hat{b})}{1+\alpha}\right) k_{t+1} + \frac{\bar{n}}{\gamma} \frac{\alpha}{1+\alpha} k_{t+1}^{1-\gamma} + \bar{n}\bar{\delta}}{1 - \gamma(1+\hat{b})} \right)^{\frac{1}{\gamma}-1} \frac{\frac{\bar{n}\alpha}{1+\alpha} (1-\gamma) k_{t+1}^{-\gamma-1}}{1 - \gamma(1+\hat{b})}, \end{aligned} $$

which simplifying boils down to

$$ \left(\frac{1}{\gamma} - 1\right) \left[ \frac{\left(1 + \frac{\alpha(1+\hat{b})}{1+\alpha}\right) k_{t+1}^{\gamma} + (1-\gamma) \frac{\bar{n}}{\gamma} \frac{\alpha}{1+\alpha}}{1 - \gamma(1+\hat{b})} \right] = \left( \frac{\left(1 + \frac{\alpha(1+\hat{b})}{1+\alpha}\right) + \frac{\bar{n}}{\gamma} \frac{\alpha}{1+\alpha} k_{t+1}^{-\gamma} + \bar{n}\bar{\delta} k_{t+1}^{-1}}{1 - \gamma(1+\hat{b})} \right) \frac{\frac{\bar{n}\alpha}{1+\alpha} (1-\gamma)}{1 - \gamma(1+\hat{b})}. \quad (29) $$

The LHS of (29) increases monotonically from $(\frac{1}{\gamma} - 1) \frac{(1-\gamma)\frac{\bar{n}}{\gamma}}{1-\gamma(1+\hat{b})} > 0$ when $k_{t+1} \to 0$ to $+\infty$ when $k_{t+1} \to +\infty$, while the RHS decreases monotonically from $+\infty$ when $k_{t+1} \to 0$ to $\frac{1+\frac{\alpha(1+\hat{b})}{1+\alpha}}{1-\gamma(1+\hat{b})} \frac{\frac{\bar{n}}{\gamma}}{1-\gamma(1+\hat{b})} > 0$ when $k_{t+1} \to +\infty$. As a consequence, the second derivative of (27) vanishes exactly once at some $\tilde{k}_{t+1} > 0$ corresponding to the unique inflection point of (27). This result combined with the limit points of (28), guarantees that (27) is concave in $k_{t+1} \in [0, \tilde{k}_{t+1})$ and convex in $k_{t+1} \in (\tilde{k}_{t+1}, +\infty)$. $\square$
Notice now that the shape of (27) depends, among the other parameters, on public spending $\bar{n}$ on healthcare. We now thus wonder how (27) shifts in the plane when $\bar{n}$ varies. In the next Proposition we show that by progressively increasing $\bar{n}$ from zero upwards, one obtains a family of maps $F^{NB}(k_{t+1}; \bar{n})$ shifting smoothly upwards in the plane. By continuity, there will then exists a unique $\hat{n} > 0$ such that $F^{NB}(k_{t+1}; \hat{n})$ is tangent to the bisector $k_t = k_{t+1}$ and lies on it.

**Proposition 2.** Under Assumption 2, there exists $\hat{n} > 0$ such that $F^{NB}(k_{t+1}; \hat{n})$ is tangent to the bisector $k_t = k_{t+1}$ and lies on it.

*Proof.* The derivative of (27) evaluated at the steady state is immediately computed as

$$ \frac{\partial F^{NB}(k_{t+1})}{\partial k_{t+1}} = \frac{1 + \frac{\alpha(1+\hat{b})}{(1+\alpha)} + \frac{\alpha\bar{n}(1-\gamma)k^{-\gamma}}{\gamma(1+\alpha)}}{\gamma k^{\gamma-1}[1 - \gamma(1+\hat{b})]} \quad (30) $$

Setting (30) equal to one and solving for $\bar{n}$, one obtains

$$ \bar{n} = \frac{\gamma(1+\alpha)}{\alpha(1-\gamma)k^{-\gamma}} \left( \gamma k^{\gamma-1}[1-\gamma(1+\hat{b})] - \left(1+\frac{\alpha(1+\hat{b})}{(1+\alpha)}\right) \right). \quad (31) $$

At the steady state, (27) writes

$$ k + \frac{\alpha[\gamma k^{\gamma}(1+\hat{b}) + \bar{n}]}{\gamma k^{\gamma-1}(1+\alpha)} + \bar{n}\bar{\delta} = k^{\gamma}[1-\gamma(1+\hat{b})]. \quad (32) $$

Plugging $\bar{n}$ obtained in (31) into (32) and rearranging terms, one has that the capital level required to make (27) tangent to the bisector, solves

$$ \begin{aligned} & \left[ (1 - \gamma(1 + \hat{b})) \frac{2\gamma - 1}{1 - \gamma} - \frac{\gamma}{1 - \gamma} \frac{1 + \alpha}{\alpha} \bar{\delta} \left( 1 + \frac{\alpha}{1 + \alpha} (1 + \hat{b}) \right) \right] k^{\gamma} + \frac{\gamma^2}{1 - \gamma} \frac{1 + \alpha}{\alpha} \bar{\delta} (1 - \gamma(1 + \hat{b})) k^{2\gamma - 1} \\ & = \left[ 1 + \frac{\alpha}{1 + \alpha} (1 + \hat{b}) \right] \frac{\gamma}{1 - \gamma} k. \end{aligned} \quad (33) $$

Under Assumptions 1 and 2, the LHS of (33) is monotonically decreasing from $+\infty$ when $k \to 0$ to $-\infty$ when $k \to +\infty$, while the RHS increases monotonically from zero when $k \to 0$ to $+\infty$ when $k \to +\infty$. As a consequence there exists a unique pair $(\hat{n}, \hat{k}(\hat{n})) > 0$ solving simultaneously (31) and (33). $\square$

We can now exploit the results included in Propositions 1 and 2, to fully characterize the existence, uniqueness, multiplicity and global stability of the stationary solutions of (27).

**Proposition 3.** Under Assumptions 1 and 2, there exists $\hat{n} > 0$ such that the equation defined by (27):

(i) for $\bar{n} \in [0, \hat{n})$, possesses a stable stationary solution $k_L$ and an unstable one $k_H$, with $k_L < k_H$;

(ii) for $\bar{n} = \hat{n}$, possesses a unique stationary solution which is stable from below and unstable from above;

(iii) for $\bar{n} > \hat{n}$, there are no stationary solutions.
*Proof.* The proof comes from coupling the features

$$F^{NB}(0) > 0, \quad \lim_{k_{t+1} \to +\infty} F^{NB}(k_{t+1}) = +\infty, \quad \lim_{k_{t+1} \to 0^+} \left( \frac{\partial F^{NB}(k_{t+1})}{\partial k_{t+1}} \right) = \lim_{k_{t+1} \to +\infty} \left( \frac{\partial F^{NB}(k_{t+1})}{\partial k_{t+1}} \right) = +\infty,$$

with the proof of the existence of the unique inflection point $\tilde{k}_{t+1}$, from the fact that $F^{NB}(k_{t+1})$ is upward shifting in the plane in $\bar{n}$, and from the definition of $\hat{n}$ in Proposition 2. It follows that for $\bar{n} > \hat{n}$, (27) must first cross the bisector at some $k_L$ such that $(\partial F^{NB}(k_{t+1}) / \partial k_{t+1})_{k_{t+1}=k_L} < 1$ and then at some $k_H$ such that $(\partial F^B(k_{t+1}) / \partial k_{t+1})_{k_{t+1}=k_H} > 1$. Moreover, for $\bar{n} = \hat{n}$ the tangent point $\hat{k}(\hat{n})$ represents the unique stationary solution, stable from below and unstable from above. $\square$

Proposition 3 claims that equation (27) generally possesses either two stationary solutions or zero, depending on whether it is placed below or above enough in the plane. When there are two stationary solutions, the lower one is stable in backward dynamics (unstable in forward one), while the highest one is unstable in backward dynamics (stable in forward one). In Figure 1, we have depicted the function (27) in red when it possesses two stationary solutions, in blue when it does not possess any stationary solution and in black when the solution is unique.

<figure><img src="image_6.png" /><figcaption>Figure 3: Function $k_t = F^{NB}(k_{t+1})$</figcaption></figure>

We can now analyze the dynamic system (26) which describes intertemporal equilibrium emerging when constraint (13) binds.

### 2.4.2 Equilibrium Dynamics: Binding Regime

When constraint (13) binds, one must set $l_t \equiv 0$ in equations (10), (11) and (12) and combine them with the FOC's (17) and (18). After some straightforward computations, we obtain the first-order difference equation for capital (26) which can be inverted to obtain the backward dynamics in explicit from

$$k_t = F^B(k_{t+1}) = \left( \frac{k_{t+1} (1 + \alpha(1 + \hat{b})) + \bar{n}\bar{\delta}}{1 - \gamma(1 + \hat{b})} \right)^{\frac{1}{\gamma}}. \quad (34)$$
Notice that under Assumption 2, (34) is always positively defined for $k_{t+1} \ge 0$. The following Proposition shows that (34), beside being positively defined, is monotonically increasing and strictly convex.

**Proposition 4.** $F^B(k_{t+1})$ is continuous, differentiable, monotonically increasing and strictly convex in $k_{t+1} \ge 0$. Moreover $F^B(0) = \left(\frac{\bar{n}\bar{\delta}}{1-\gamma(1+\hat{b})}\right)^{\frac{1}{\gamma}} > 0$ and $\lim_{k_{t+1} \to +\infty} F^B(k_{t+1}) = +\infty$.

*Proof.* Continuity, differentiability and limit points are trivially verified. Monotonicity and strictly convexity are guaranteed by the positive sign of, respectively, the first and second derivatives of $F^B(k_{t+1})$. In fact, for all $k_{t+1} \ge 0$ one has

$$ \frac{\partial F^B(k_{t+1})}{\partial k_{t+1}} = \frac{1}{\gamma} \left( \frac{k_{t+1} \left(1 + \alpha(1 + \hat{b})\right) + \bar{n}\bar{\delta}}{1 - \gamma(1 + \hat{b})} \right)^{\frac{1-\gamma}{\gamma}} \frac{1 + \alpha(1 + \hat{b})}{1 - \gamma(1 + \hat{b})} > 0 \quad (35) $$

and

$$ \frac{\partial^2 F^B(k_{t+1})}{\partial k_{t+1}^2} = \frac{1-\gamma}{\gamma} \frac{1}{\gamma} \left( \frac{k_{t+1} \left(1 + \alpha(1 + \hat{b})\right) + \bar{n}\bar{\delta}}{1 - \gamma(1 + \hat{b})} \right)^{\frac{1-2\gamma}{\gamma}} \left( \frac{1 + \alpha(1 + \hat{b})}{1 - \gamma(1 + \hat{b})} \right)^2 > 0. \quad (36) $$

□

Looking at equation (34), one has that for $\bar{n} = 0$, it passes through the origin of the axes and, as $\bar{n}$ is progressively increased, it undergoes upward shifts in the plane. This means that for $\bar{n}'$s low enough, (34) will intersect the bisector $k_t = k_{t+1}$ twice; then, in correspondence to some critical $\hat{n}$, exactly once. Finally, for $\bar{n} > \hat{n}$, there will no longer be any intersection. In the next Proposition we provide the analytical expression of such critical $\hat{n}$.

**Proposition 5.** There exists a unique $\hat{n} > 0$ such that $F^B(k_{t+1})$ is tangent to the bisector and lies on it.

*Proof.* Setting the first derivative (35), evaluated at the steady state, equal to one, and by rearranging terms one obtains

$$ \bar{n} = \frac{\left(\frac{\gamma}{1+\alpha(1+\hat{b})}\right)^{\frac{\gamma}{1-\gamma}} \left(1-\gamma(1+\hat{b})\right)^{\frac{1}{1-\gamma}} - \left(1+\alpha(1+\hat{b})\right)k}{\bar{\delta}}. \quad (37) $$

Plugging (37) into (34) evaluated at the steady state, one gets

$$ (1 - \gamma(1 + \hat{b})) k^{\gamma} = \left( \frac{\gamma}{1 + \alpha(1 + \hat{b})} \right)^{\frac{\gamma}{1-\gamma}} (1 - \gamma(1 + \hat{b}))^{\frac{1}{1-\gamma}}. \quad (38) $$

The LHS of (38) increases monotonically from zero ($k = 0$) to $+\infty$ ($k \to +\infty$), while the RHS of (37) is a positive constant. Then there exists a unique $\bar{k} > 0$ solving (38) and thus a unique $\hat{n}(\bar{k}) > 0$ solving (37). The expressions of $\bar{k}$ and $\hat{n}$ are, respectively,

$$ \bar{k} = \left( \frac{\gamma \left(1 - \gamma(1 + \hat{b})\right)}{1 + \alpha(1 + \hat{b})} \right)^{\frac{1}{1-\gamma}} $$
and

$$ \hat{n} = \frac{\left( \frac{\gamma}{1+\alpha(1+\hat{b})} \right)^{\frac{\gamma}{1-\gamma}} \left( 1 - \gamma(1+\hat{b}) \right)^{\frac{1}{1-\gamma}} - \left( 1 + \alpha(1+\hat{b}) \right) \bar{k}}{\bar{\delta}}. $$

□

In light of Proposition 4, we can now fully characterize the existence, multiplicity and stability of the stationary solutions of (27) as $\bar{n}$ is made to increase from zero on.

**Proposition 6.** Under Assumptions 1 and 2, there exists $\hat{n} > 0$ such that the equation defined by (27):

(i) for $\bar{n} \in [0, \hat{n})$, possesses a globally unstable stationary solution $k_L$ and a globally stable one $k_H$, with $k_L < k_H$;

(ii) for $\bar{n} = \hat{n}$, possesses a unique stationary solution which is globally unstable from below and globally stable from above;

(iii) for $\bar{n} > \hat{n}$, there are no stationary solutions.

*Proof.* The existence and multiplicity of the stationary solutions comes from

$$ F^B(0) > 0, \quad \lim_{k_{t+1} \to +\infty} F^B(k_{t+1}) = +\infty, \quad \lim_{k_{t+1} \to 0^+} \left( \frac{\partial F^{NB}(k_{t+1})}{\partial k_{t+1}} \right) = \lim_{k_{t+1} \to +\infty} \left( \frac{\partial F^{NB}(k_{t+1})}{\partial k_{t+1}} \right) = +\infty, $$

from the fact that $F^{NB}(k_{t+1})$ is upward shifting in $\bar{n}$ and from the definition of $\hat{n}$. As a consequence, (27) must first cross the bisector at some $k_L$ such that $\left( \frac{\partial F^B(k_{t+1})}{\partial k_{t+1}} \right)_{k_{t+1}=k_L} < 1$ and then at some $k_H$ such that $\left( \frac{\partial F^B(k_{t+1})}{\partial k_{t+1}} \right)_{k_{t+1}=k_H} > 1$. □

In Figure 4, we have plotted three curves corresponding to the cases (i), (ii), and (iii) of Proposition 6, respectively. The red curve exhibits two stationary solutions, the blue line a unique one, while the yellow one does not possess any fixed point.

Up to this point, we have characterized in a separate way equations (34) and (27) describing, in backward dynamics, intertemporal equilibrium corresponding to a binding constraint (13) and to a non-binding one, respectively. Since constraint (27) binds if and only if $k_t < (23)$, we can define the complete equilibrium dynamics of the economy as well as characterize its transition from the binding regime for constraint (13) to the non-binding one.

### 2.4.3 Global Dynamics

We have seen that the backward dynamics of intertemporal equilibrium is governed by equation (34) when constraint (13) binds, and by equation (27) when it does not bind. We have also shown that constraint (13) is binding if and only if (23) holds. In order to analyze the complete equilibrium dynamics obtained by combining the two regimes, the following Proposition includes an useful result establishing under which conditions equations (27) and (34) evaluated at $k_{t+1} = k^{TH}$ lie under or below the bisector $k_t = k_{t+1}$.

**Proposition 7.** $F^B(k^{TH}) = F^{NB}(k^{TH}) > k^{TH}$ if and only if

$$ \bar{n} > \left[ \frac{[1 - \gamma(1 + \hat{b})(1 + \delta\alpha)]}{(1 + \alpha(1 + \hat{b}))} \right]^{\frac{\gamma}{1-\gamma}} \alpha\gamma(1 + \hat{b}) \equiv \bar{n}^{TH}. \quad (39) $$
<figure><img src="image_7.png" /><figcaption>Figure 4: Function $k_t = F^B(k_{t+1})$</figcaption></figure>

Proof. Condition $F^B(k^{TH}) > k^{TH}$ writes

$$F^B(k^{TH}) = \left[ \frac{\left(1 + \alpha(1 + \hat{b})\right) \left(\frac{\bar{n}}{\alpha\gamma(1+\hat{b})}\right)^{\frac{1}{\gamma}} + \bar{n}\bar{\delta}}{[1 - \gamma(1 + \hat{b})]} \right]^{\frac{1}{\gamma}} > \left(\frac{\bar{n}}{\alpha\gamma(1 + \hat{b})}\right)^{\frac{1}{\gamma}}.$$

Removing the exponent and rearranging terms, we get

$$\bar{n} \left[ \left( 1 + \alpha(1 + \hat{b}) \right) \left[ \frac{\bar{n}}{\alpha\gamma(1 + \hat{b})} \right]^{\frac{1}{\gamma} - 1} \left[ \frac{1}{\alpha\gamma(1 + \hat{b})} \right] + \bar{\delta} \right] > \left[ \frac{\bar{n}}{\alpha\gamma(1 + \hat{b})} \right] [1 - \gamma(1 + \hat{b})].$$

Solving for $\bar{n}$, one thus obtains expression (39). □

Note that in the special case $\gamma(1 + \hat{b})(1 + \delta\alpha) > 1$ (which is more likely to be verified when $\hat{b}$, $\delta$ and $\alpha$ are large) inequality (39) holds for every $\bar{n}$. The next Proposition allows us to rank the threshold $k^{TH}$ with respect to the stationary solutions of equations $F^B$ and $F^{NB}$, provided that such stationary solutions exist i.e., in view of Propositions 1 and 2, $\bar{n}$ is not too large. In particular, denoting $k_L^B$ and $k_H^B$ the low and the high steady state of $F^B$, and $k_L^{NB}$ and $k_H^{NB}$ the low and the high steady state of $F^{NB}$, we prove that when (39) holds, one has $k^{TH} < k_L^{NB} < k_L^B < k_H^B < k_H^{NB}$; otherwise, one gets $k_L^B < k_L^{NB} < k^{TH} < k_H^B < k_H^{NB}$.

**Proposition 8.** Assume that both $F^B(k)$ and $F^{NB}(k)$ possess two stationary solutions denoted by $k_L^B$ and $k_H^B$, with $k_L^B < k_H^B$, and $k_L^{NB}$ and $k_H^{NB}$, with $k_L^{NB} < k_H^{NB}$. Then

(i) when (39) holds, one has $k^{TH} < k_L^{NB} < k_L^B < k_H^B < k_H^{NB}$;

(ii) when (39) does not hold, one gets $k_L^B < k_L^{NB} < k^{TH} < k_H^B < k_H^{NB}$.

*Proof.* To prove that we are either in regime (i) or (ii), we start by fixing $\bar{n}$ arbitrarily close to zero. In such a case, also $k^{TH}$ will be very close to zero, as well as the inflection point $\tilde{k}_{t+1}$
for $F^{NB}$. Since in this case $k^{TH}$ will lie above the bisector, the ranking in (i) is immediately verified.

As soon as $\bar{n}$ is slightly relaxed, the inflection point occurs for larger capital levels, while both $F^B$ and $F^{NB}$ shift upward. As a consequence, $F^B$ and $F^{NB}$ will cross the bisector at some $k_L^B$ and $k_L^{NB}$ satisfying $k_L^B < k_L^{NB} < k^{TH}$. Similarly, $F^B$ and $F^{NB}$ will cross for a second time the bisector at, respectively, some $k_H^B$ and $k_H^{NB}$ satisfying $k^{TH} < k_H^B < k_H^{NB}$.

Finally, by further increasing $\bar{n}$, ranking (ii) will still hold until $\bar{n}$ is large enough to rule out the existence of stationary solutions. □

Up to this point, we have studied functions (27) and (34) describing the backward dynamics of the different regimes, namely the binding and non-binding ones. However, it is straightforward to translate all the previous results in terms of forward dynamics. To this end, the next Proposition characterizes the global dynamics of the economy together with the transition from one regime to the other. In light of Proposition 8, there may indeed emerge two types of global dynamics, depending on whether public health supply $\bar{n}$ satisfies condition (39).

**Proposition 9.** Assume that both $F^B$ and $F^{NB}$ possess two stationary solutions. Then

(i) when (39) holds, then equilibrium dynamics is described by (34) for $0 < k_t < k^{TH} < k_L^{NB} < k_L^B < k_H^B < k_H^{NB}$ and by (27) for $k_t > k^{TH}$;

(ii) when (39) does not hold, then equilibrium dynamics is described by (34) for $k_t < k^{TH} < k_H^B < k_H^{NB}$ and by (27) for $k_t > k^{TH}$.

*Proof.* It comes directly from Proposition 8 applied to the forward dynamics. □

In Figures 5 and 6, we have plotted the global forward dynamics of the economy described by equation $k_{t+1} = F(k_t)$ encompassing cases (i) and (ii) of Proposition 9, and defined as

$$k_{t+1} = F(k_t) = \begin{cases} (F^{NB})^{-1}(k_{t+1}) & \text{for } k_{t+1} \le k^{TH} \\ (F^B)^{-1}(k_{t+1}) & \text{for } k_{t+1} > k^{TH} \end{cases} \quad (40)$$

where $(F^{NB})^{-1}$ and $(F^B)^{-1}$ stand for the inverse of the functions $F^{NB}$ and $F^B$, respectively.

In Figures 5 and 6 we can see that the function $F(k_t)$ is initially convex, then undergoes an inflection point and finally becomes concave. In Figure 5, which corresponds to case (i) of Proposition 9, both the effective stationary solutions fall within the non-binding regime. Figure 6 instead refers to case (ii) of Proposition 8. The relevant low stationary state will therefore correspond to that of the binding regime, while the high one to that of the non-binding one.
<figure><img src="image_8.png" /><figcaption>Figure 5: Forward dynamics $k_{t+1} = F(k_t)$ for case (i)</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Figure 6: Forward dynamics $k_{t+1} = F(k_t)$ for case (ii)</figcaption></figure>

The previous Proposition characterizes the global dynamics of the market economy and proves that two capital stationary solutions exist, the low one being globally unstable, the high one stable. However, in Propositions 3 and 6, we had proved that the existence of the two stationary solutions requires a level of public healthcare $\bar{n}$ not large enough. In fact, as the latter increases from zero on, the two stationary solutions begin to get closer and closer until they merge at a certain critical threshold $\tilde{n}$ and then definitively disappear. In Figure 7, we have depicted
the phase diagram of the economy in which the two stationary states are plotted as a function of $\bar{n}$. The arrows indicate the global stability of the stationary solutions: as $\bar{n}$ increases, the two solutions get closer, although their stability features persist, except when they coincide. At that point, in fact, the unique stationary solution becomes stable from above and unstable from below.

<figure><img src="image_10.png" /><figcaption>Figure 7: Phase Diagram</figcaption></figure>

## 2.5 Time Trends

We have established that, provided public healthcare supply $\bar{n}$ is not too large, there emerge two stationary solutions, the high one being globally stable while the low one globally unstable. Being the dynamics governing capital accumulation one-dimensional, it is quite simple to linearize it and find the explicit trajectory of capital as a function of time.

To this end, we adopt the following calibration for the structural parameters: $\gamma = 0.33$, $\alpha = 1$, $\hat{b} = 0.1$, $\bar{n} = 0.1$. Note that these values fall within empirically plausible ranges and give rise to the stationary solutions $k_L = 0.0005$ and $k_H = 0.1654$. The threshold is immediately computed as $k^{TH} = 0.0201$ and satisfies $k_L < k^{TH} < k_H$. As a consequence $k_L$ falls within the regime corresponding to a binding constraint (13) while $k_H$ within that corresponding to a non-binding one. By linearizing (26) around $k_H$, the dynamics of the deviation from the steady state, $dk_t = k_t - k_H$, is

$$dk_{t+1} = J_k dk_t,$$

where $J_k = \frac{(1+\alpha)[1-\gamma(1+\hat{b})]\gamma^2(k_H)^{\gamma-1}}{\gamma[1+\alpha(2+\hat{b})]+\alpha\bar{n}(1-\gamma)(k_H)^{-\gamma}}$. We will then have the following time trend of $dk_t$ for a given initial condition $dk_0 = k_0 - k_H$:

$$dk_{t+1} = J_k dk_0. \qquad (41)$$

Since from the FOC's we obtain the expressions of all the endogenous variables in terms of capital, it is immediate to derive their time trend. The dynamics of GDP will in fact be

$$dy_t = J_y J_k dk_0 \qquad (42)$$

where $J_y = \gamma k_H^{\gamma-1}$, the one linked to consumption

$$dc_t = J_c J_k dk_0 \qquad (43)$$
with $J_c = \frac{(1+\hat{b})\gamma 2k_H^{\gamma-1}}{1+\alpha}$, that of private healthcare

$$dl_t = J_l J_k dk_0 \quad (44)$$

where $J_l = \frac{(1+\hat{b})\alpha\gamma 2k_H^{\gamma-1}}{1+\alpha}$, that of investment in preventive medicine reads

$$dA_t = J_A J_k dk_0 \quad (45)$$

where $J_A = \frac{\alpha\phi(\gamma(1+\hat{b})+(1-\gamma)\bar{n}k_H^{\gamma-1})}{\gamma(1+\alpha)}$ and finally that of the endogenous component of waiting times

$$d\beta_t = J_\beta J_k dk_0 \quad (46)$$

with $J_\beta = -\frac{(1+\alpha)(1+\hat{b})\gamma^2 k_H^{\gamma-1}}{\bar{n}[(1+\hat{b})\gamma k_H^{\gamma-1}+\bar{n}]^2}$. In Figure 8, we plot the time trends of the aforementioned variables by setting $dk_0 = -1$. As it emerges in Figure 8, all variables appear to be procyclical, except waiting times $\beta$. This should not be surprising since, when income increases, individuals have economic tools to pay for private health services to avoid excessively long waiting times, easing the pressure on public healthcare. Finally, as emerges in Figure 8, the dynamic path of GDP lies below the other procyclical variables paths, since such variables represent a fraction of GDP.

<figure><img src="image_11.png" /><figcaption>Figure 8: Time trends</figcaption></figure>

We have therefore analyzed the equilibrium dynamics of the market economy. Recall to mind that OLG economies generically are not Pareto efficient due to the failure of each generation to consider the impact of its choices on future ones. In our case, an additional source of inefficiency is related to the provision of public healthcare, assumed to be constant. To identify its optimal provision, we should maximize aggregate welfare defined over the infinite sequence of generations. Actually, in the next Section, we characterize the optimal intertemporal allocation of resources chosen by a benevolent social planner who cares about the welfare of all generations, although she discounts those more distant in the future.
### 3 Aggregate Welfare and Optimality

In this Section we analyze the optimal intertemporal allocation problem solved by a social planner who cares for the welfare of all present and future generations. The solution to this problem allows us to identify the set of Pareto efficient allocations, i.e. the contract curve. Specifically, the social planner discounts future generations' welfare at the factor $\rho \in (0, 1)$ capturing the degree of collective "patience" and ranging from zero (maximum impatience) to one (maximum patience). We will see how, not surprisingly, the social planner's solution varies dramatically as $\rho$ is continuously relaxed from zero to one. In view of these considerations, the social planner's problem consists of maximizing the infinite stream of instantaneous utilities of individuals belonging to different generations, weighting each utility by the discount factor $\rho$. Note that the social planner does not operate within a market economy; indeed, her problem consists in choosing how to allocate resources given agents' preferences and technological possibilities. For these reasons, in particular, at each date $t$ public healthcare consumption $n_t$ is not set by the government as in the market economy, but is the direct outcome of the social planner's maximization program.

However, as we will explain in detail, public healthcare is not a perfect substitute for private healthcare for two main reasons. First, as we have already seen, it entails a (quadratic) disutility by virtue of a supposed technology that requires waiting times, reducing the efficiency and performance of the public healthcare sector. We frame such a disutility related to public healthcare consumption $n_t$ as $\beta \frac{1}{2} n_t^2$, where $\beta > 0$ is a parameter now fixed constant and exogenous, as there is no longer any need for a variable disutility coefficient associated to waiting times. Moreover, one has not to adjust $\beta$ to match the demand for public healthcare to its supply. The instantaneous utility function of the generation born in $t$ is therefore

$$U(c_t, l_t, n_t) = \ln c_t + \alpha \ln (A_t(l_t + n_t)) - \beta \frac{1}{2} n_t^2. \quad (47)$$

Second, we again assume that the efficient units of public healthcare consumption are larger than those of private healthcare consumption. In particular, a unit of public healthcare $n$ can be transformed into $\delta < 1$ unit of private healthcare $l$, according to Assumption 1. In other words, $\delta$ represents the (constant) marginal rate of transformation between public and private healthcare and is still assumed to be lower than one to ensure a positive solution for public healthcare, preventing it to be otherwise driven to zero. The intertemporal utility that the social planner must maximize is therefore

$$\sum_{t=0}^{\infty} \rho^t U(c_t, l_t, n_t) = \sum_{t=0}^{\infty} \rho^t \left( \ln c_t + \alpha \ln (A_t(l_t + n_t)) - \beta \frac{1}{2} n_t^2 \right), \quad (48)$$

where all the variables $c_t$, $l_t$, $n_t$, $A_t$, are the same as those introduced in the decentralized economy. Since the social planner is concerned only with the optimal and technologically feasible allocations of resources, in each period $t$ she is subject only to the aggregate resources constraint

$$k_{t+1} + c_t + l_t + \delta n_t + I_t = k_t^\gamma \quad (49)$$

and to the technological relationship between investment $I_t$ in preventive medicine in time $t$ and health status $A_{t+1}$ in $t+1$

$$A_{t+1} = \phi I_t. \quad (50)$$

Finally, the non-negativity constraint on private health consumption must be introduced, i.e.

$$l_t \ge 0. \quad (51)$$
As usual, the transversality condition must be respected, in order to avoid capital over-accumulation or Ponzi games:

$$ \lim_{t \to \infty} \rho^t (c_t^{-1} + \alpha A_t^{-1} + \alpha (l_t + n_t)^{-1} (1 - \beta n_t)) k_t = 0. \quad (52) $$

Under the hypothesis $\delta < 1$, we will prove that public health will always be strictly positive, even if it becomes smaller and smaller as the private one grows. The social planner's problem thus consists in choosing an infinite sequence $\{k_t, c_t, l_t, n_t, A_t, I_t\}_{t=0}^{\infty}$ with the aim of maximizing (47) under constraints (49), (50) and (51) and respecting (9). Note that the two state variables, predetermined in period zero, are $k_0 > 0$ and $I_{-1} > 0$ (and therefore $A_0 > 0$).

Plugging the expression for $A_{t+1}$ in (50) into (47), and assuming that constraint (51) is non-binding, after some manipulations we obtain the following FOCs:

$$ c_{t+1} = \rho c_t \gamma k_{t+1}^{\gamma-1}, \quad (53) $$

$$ I_t = \rho \alpha c_t, \quad (54) $$

$$ n_t = \frac{1 - \bar{\delta}}{\beta c_t} \quad (55) $$

and

$$ l_t = \alpha c_t - \frac{1 - \bar{\delta}}{\beta c_t}. \quad (56) $$

Equation (53) is the standard Euler one establishing the optimal consumption smoothing behavior, while (54), (55) and (56) are optimal arbitrage conditions between consumption and investment in preventive medicine, public health and private one, respectively.

If instead constraint (50) binds, the whole intertemporal optimization problem must be reformulated by setting $l_t \equiv 0$ into constraints (47) and (49). By computing and manipulating the FOC's, one obtains, besides the Euler equation (53), the alternative arbitrage condition between consumption and public health

$$ c_t = \frac{\bar{\delta} n_t}{\alpha - \beta n_t^2} \quad (57) $$

and that between investment in private medicine and public health

$$ I_t = \rho \alpha \frac{\bar{\delta} n_t}{\alpha - \beta n_t^2}. \quad (58) $$

We can now combine the FOCs with the resources constraint (49) to fully characterize the social planner's solution. To this end, we begin by assuming that constraint (51) is non-binding. From (56), this is true if and only if

$$ c_t > \sqrt{\frac{1 - \bar{\delta}}{\alpha \beta}} \equiv c^{TH}. \quad (59) $$

Under the domain of validity of (59), we can substitute the expressions (54), (55) and (56) into the resources constraint (49) to obtain the dynamic equation of capital accumulation as a function of consumption $c_t$

$$ k_{t+1} + (1 + \alpha(1 + \rho))c_t - \frac{(1 - \bar{\delta})^2}{\beta c_t} = k_t^{\gamma}. \quad (60) $$

As a consequence, under a non-binding constraint (51), the social planner's solution is represented by a sequence $\{k_t, c_t\}_{t=0}^{\infty}$, with $k_0 > 0$ given, satisfying simultaneously equations (53) and (60) as well as the transversality condition (52).
By contrast, when constraint (51) is binding, one should set $l_t = 0$ for every $t$ and plug the expression (57) for consumption and the expression (58) for investment in preventive medicine into the resources constraint (49), to obtain the dynamic equation of capital accumulation in terms of public healthcare $n_t$

$$k_{t+1} + \left( \frac{1 + \alpha(1 + \rho) - \beta n_t^2}{\alpha - \beta n_t^2} \right) \bar{\delta} n_t = k_t^{\gamma}. \quad (61)$$

As a result, as soon as constraint (51) binds (i.e. $c_t < c^{TH}$), the optimal intertemporal allocation chosen by the social planner will be represented by a sequence $\{k_t, n_t\}_{t=0}^{\infty}$ satisfying equations (53) and (61) as well as the transversality condition (52). Note that, as usual, capital represents a predetermined variable whose initial value $k_0$ is known, while initial consumption $c_0$ and public healthcare $n_0$ are jump variables, i.e. non-predetermined and therefore the social planner must choose their initial values in order to respect the transversality condition (52). In turn, the fulfilment of (52) requires the convergence of the sequences $\{k_t, c_t\}_{t=0}^{\infty}$ and $\{k_t, n_t\}_{t=0}^{\infty}$ toward the stationary solution or toward some other attractor. The next Proposition characterizes the social planner intertemporal allocations both under a binding and a non-binding constraint (51).

**Proposition 10.** The intertemporal solution of the social planner optimization problem is a sequence $\{k_t, c_t, n_t, I_t, A_t, l_t\}_{t=0}^{\infty}$, with $k_0 > 0$ and $I_{-1}$ given, solving equations (50), (53), (57), (58), (61), $l_t \equiv 0$ and (52) when $c_t < c^{TH}$, and equations (50), (52), (53), (54), (55), (56) and (60) when $c_t > c^{TH}$.

Having derived the equilibrium dynamics of the planned economy, we move on to analyze the existence and uniqueness of its stationary solution. When constraint (51) is non-binding, it is easy to verify from the Euler equation (53) evaluated at the steady state that the unique level of the stationary capital $k^{SP}$ depends exclusively of the technological parameter $\gamma$ and the discount factor $\rho$. In fact, one has

$$k^{SP} = (\rho\gamma)^{\frac{1}{1-\gamma}}. \quad (62)$$

It follows that $k^{SP}$ is strictly lower than one and increasing in $\rho \in (0, 1)$. Once the unique stationary value of capital, $k^{SP}$, has been found, the corresponding stationary value for consumption $c^{SP}$ is obtained by solving for $c$ the expression

$$c(1 + \alpha(1 + \rho)) - \frac{(1 - \bar{\delta})^2}{\beta c} = (k^{SP})^{\gamma} - k^{SP}. \quad (63)$$

Noting that $(k^{SP})^{\gamma} - k^{SP} > 0$ and that the LHS of (63) increases monotonically from $-\infty$ when $c \to 0$ to $+\infty$ when $c \to +\infty$, we have proved that there exists a unique stationary consumption level $c^{SP} > 0$. However, since these results are found under the assumption that constraint (51) is non-binding and this in turn requires consumption to be larger than $c^{TH}$ defined in (59), the stationary consumption $c^{SP}$ obtained from (63) is consistent with the assumption of a non-binding constraint (51) evaluated at the steady state if and only if it satisfies $c^{SP} > c^{TH}$. Once one observes that $(k^{SP})^{\gamma} - k^{SP}$ is decreasing in $\rho$ and thus $c^{SP}$ is increasing, we can claim that a steady state $(k^{SP}, c^{SP})$ is consistent with the assumption of a non-binding constraint (51) if and only if the discount factor $\rho$ is larger than the critical value $\hat{\rho}$ set in order to make the associated stationary capital $\hat{k} = (\hat{\rho}\gamma)^{\frac{1}{1-\gamma}}$ to satisfy

$$c^{TH} (1 + \alpha (1 + \rho)) - \frac{(1 - \bar{\delta})^2}{\beta c^{TH}} = (\hat{\rho}\gamma)^{\frac{\gamma}{1-\gamma}} - (\hat{\rho}\gamma)^{\frac{1}{1-\gamma}}. \quad (64)$$

In light of all these considerations, the stationary consumption $c^{SP}$, as well as the stationary investment $I^{SP}$ in preventive medicine and private healthcare $l^{SP}$ increase with the degree of
collective patience $\rho$, along with stationary capital. Conversely, public healthcare consumption $n^{SP}$ decreases with $\rho$, as is clear from checking equation (55): faced with an increase in income and therefore in private healthcare, the marginal utility of public healthcare consumption decreases faster than its marginal disutility and thus the social planner will reduce its allocation.

When constraint (51) is by contrast binding, the stationary capital is always obtained from the equation (62) while the stationary consumption is obtained by solving for $n$ equation (61) evaluated at steady state, i.e.

$$ \bar{\delta}n + (1 + \alpha) \left( \frac{\bar{\delta}n}{\alpha - \beta n^2} \right) = (k^{SP})^{\gamma} - k^{SP}. \quad (65) $$

Always in light of the fact that $(k^{SP})^{\gamma} - k^{SP} > 0$ and observing that the LHS of (65) increases monotonically from zero to infinite when $n$ moves from zero to $\sqrt{\frac{\alpha}{\beta}}$, we have a unique solution $n^{SP}$ included in $(0, \sqrt{\frac{\alpha}{\beta}})$. The next Proposition, immediately proved, ensures the existence and uniqueness of the stationary solution for the social planner's problem both when constraint (51) binds and when it does not.

**Proposition 11.** The social planner optimization problem possesses an unique stationary solution. When $\rho < \hat{\rho}$ the solution $(k^{SP}, n^{SP})$ is obtained from (62) and (65), while for $\rho > \hat{\rho}$ the solution $(k^{SP}, c^{SP})$ is obtained from (62) and (63).

We now study the stability of the stationary solution of the dynamic systems defined either by the equations (53) and (60) or by equations (53) and (61). Since among the two variables involved, capital is a predetermined one while ordinary consumption $c$ as well as public health consumption $n$ are non-predeterminate, the (local) uniqueness of the stationary equilibrium exhibits the saddle path stability. To prove this, we first consider the non-binding regime, i.e. $\rho > \rho^{TH}$, described by equations (53) and (60). The linearized dynamics in terms of the deviations $dk_t$ and $dc_t$ from their stationary values is given by

$$ \begin{bmatrix} dk_{t+1} \\ dc_{t+1} \end{bmatrix} = J_1 \begin{bmatrix} dk_t \\ dc_t \end{bmatrix} $$

where

$$ J_1 = \begin{bmatrix} \rho^{-1} & -\mathcal{Z} \\ -\frac{1}{\rho}c^{SP}(1-\gamma)(\rho\gamma)^{\frac{1}{1-\gamma}} & 1 + \mathcal{Z}c^{SP}(1-\gamma)(\rho\gamma)^{\frac{1}{1-\gamma}} \end{bmatrix} $$

denotes the Jacobian evaluated at the steady state and $\mathcal{Z} \equiv 1 + \alpha(1 + \rho) + \frac{(1-\bar{\delta})^2}{\beta(c^{SP})^2}$. Simple computations yield the following expressions for the determinant $D$ and the trace $T$ of $J_1$

$$ D = \rho^{-1} $$

and

$$ T = 1 + D + \Lambda \quad (66) $$

where $\Lambda \equiv \mathcal{Z}c(1-\gamma)(\rho\gamma)^{\frac{1}{1-\gamma}} > 0$. The characteristic polynomial, whose roots correspond to the eigenvalues of $J_1$, is thus $P(\lambda) = \lambda^2 - (1 + D + \Lambda)\lambda + D = 0$. Notice that $P(\lambda)$ is continuous and defined on a connected set, with $\lim_{\lambda \to -\infty} P(\lambda) = \lim_{\lambda \to +\infty} P(\lambda) = +\infty$, $P(0) = D > 0$ and $P(1) = -\Lambda < 0$. We have thus proved that one root of $P(\lambda)$ is real, positive and lower than one, the other is still real and positive but larger than one. As a consequence the steady state of the system defined by (53) and (60) is saddle path stable.
Examining now the stability of the stationary solution obtained in the case $\rho < \rho^{TH}$ corresponding to a binding constraint (51), the linearized dynamics of the deviations $dk$ and $dn$ is generated by the system

$$ \begin{bmatrix} dk_{t+1} \\ dn_{t+1} \end{bmatrix} = J_2 \begin{bmatrix} dk_t \\ dn_t \end{bmatrix} $$

where

$$ J_2 = \begin{bmatrix} \rho^{-1} & -A \\ B\rho^{-1} & 1 - AB \end{bmatrix} $$

denotes the Jacobian matrix evaluated at the steady state under study, $A = \bar{\delta} + (1+\alpha)\bar{\delta}(\frac{\alpha-\beta n^2}{\alpha-\beta n^2})$ and $B = \frac{(\frac{\bar{\delta}n}{\alpha-\beta n^2}\rho\gamma(\gamma-1)k^{\gamma-2})}{(\frac{\bar{\delta}(\alpha-\beta n^2)+2\beta n^2}{\alpha-\beta n^2})}$. The determinant $D$ and the trace $T$ of $J_2$ are given by $D = \rho^{-1}$ and $T = 1+D+AB$. Thus, applying the same procedure as in the case $\rho > \rho^{TH}$, we obtain that the steady state is still saddle path stable. The following proposition, already proved, summarizes the stability properties of the planned economy.

**Proposition 12.** *The unique steady state of the social planner intertemporal allocation is saddle path stable.*

Proposition 12 claims that the unique steady state of the planned economy is saddle path stable. However, in light of the considerations made above, we have seen that this steady state may correspond to the non-binding or to the binding regime of constraint (51). In the first case, the non-binding condition will be verified for every $c_t > c^{TH}$, where $c^{TH}$ is defined in (59) and to which corresponds a stock of capital $k^{TH}$ obtained solving (60) for $k$ once one has set $c_t = c^{TH}$. If we move to the linearized system describing the dynamics of the deviations $dk_t$ and $dc_t$, the critical thresholds become thus $dk^{TH} = k^{TH} - k^{SP} < 0$ and $dc^{TH} = c^{TH} - c^{SP} < 0$. Consequently, for $dk_t < dk^{TH}$, the dynamics will be governed by the Jacobian $J_2$ while for $dk_t > dk^{TH}$ by the Jacobian $J_1$. It follows that for any initial condition $dk_0 < dk^{TH}$, in light of the saddle path stability (feature of the stationary state), the economy will move along the stable branch corresponding to the binding regime up to $dk^{TH}$. Subsequently, for $dk_t > dk^{TH}$, the system will jump to the stable branch corresponding to the non-binding regime until it will converge to the stationary state.

In Figure 9, we have plotted the complete phase diagram assuming that the steady state (and hence, in terms of deviations, the origin of the axes) corresponds to the non-binding regime for constraint (51).

In Figure 9, the solid blue line corresponds to the stable manifold $V_{NB}^S$ of the non-binding regime, while the dashed blue line corresponds to the unstable manifold $V_{NB}^{US}$. Similarly, the solid red line and the dashed red line correspond to the stable and unstable manifolds $V_B^S$ and $V_B^{US}$ of the binding regime, respectively. It is easily verifiable that the slope of the stable manifold $V_{NB}^S$ associated with the non-binding regime is larger than the slope of the stable manifold $V_B^S$ related to the binding regime, as depicted in Figure 9. Assuming as in Figure 9 that the initial condition $dk_0$ satisfies $dk_0 < dk^{TH}$, we have that consumption $dc_0$ initially jumps onto the stable manifold $V_B^S$ of the binding regime (solid red line) and begins to move along it smoothly, as the arrows indicate. However, when $dk_t$ crosses $dk^{TH}$, the economy will enter in the non-binding regime. As a consequence, consumption will undergo a sharp contraction, until it settles on the stable manifold $V_{NB}^S$ associated with the non-binding regime, as the arrows in Figure 9 again show. From that moment on, the system will converge smoothly to the stationary state, along the stable manifold $V_{NB}^S$.

The intuition underlying the sharp contraction in consumption observed when the economy moves from the binding to the non-binding regime is easily understandable. Indeed, when the regime change occurs, private healthcare goes from being identically zero to having positive
<figure><img src="image_12.png" /><figcaption>Figure 9: Phase diagram. Non-binding regime</figcaption></figure>

values that increase with capital. However, a private healthcare that suddenly becomes positive must require to sacrifice a larger amount of consumption. From a geometric point of view, this corresponds to the sharp contraction in consumption, that in Figure 9 is represented by the discrete jump moving the economy from the stable manifold $V_B^S$ to $V_{NB}^S$ in correspondence to $dk_t = dk^{TH}$.

The next Figure 10 refers to the global phase diagram emerging when the stationary solution falls conversely within the binding regime for constraint (51). Recall to mind that under this configuration, the critical thresholds satisfy $dk^{TH} = k^{TH} - k^{SP} > 0$ and $dc^{TH} = c^{TH} - c^{SP} > 0$ as it is depicted in Figure 10.

<figure><img src="image_13.png" /><figcaption>Figure 10: Phase diagram. Binding regime</figcaption></figure>
In Figure 10, the solid blue line again represents the stable manifold $V_{NB}^S$ of the non-binding regime, while the dashed blue line the unstable one, $V_{NB}^{US}$. As before, the solid red line and the dashed red line correspond instead to the stable and unstable manifolds $V_B^S$ and $V_B^{US}$ of the binding regime, respectively. Here again the slope of the stable manifold $V_{NB}^S$ is larger than the slope of the stable manifold $V_B^S$. If the initial condition $dk_0$ is such that $dk_0 > dk^{TH}$, initial consumption $dc_0$ will instantaneously jump onto the stable manifold $V_{NB}^S$ of the non-binding regime (solid blue line); the system will thus start to move smoothly toward the steady state along $V_{NB}^S$ (see the arrows in Figure 10). However, when $dk_t$ crosses $dk^{TH}$ from above, constraint (51) will start to bind. As a consequence, consumption will undergo a sharp contraction, namely it will jump below on the stable manifold $V_B^S$ associated with the non-binding regime. From that point on, the whole economy will contract smoothly toward to the stationary state, along the stable manifold $V_B^S$.

The explanation behind the sharp contraction in consumption, observed in correspondence to the regime shift occurring when $dk_t$ passes through $dk^{TH}$, is again rather intuitive. Indeed, when $dk_t > dk^{TH}$, the social planner reacts to the drop in GDP by simultaneously reducing both consumption and private healthcare. However, when constraint (51) begins to bind, the further drop in GDP will be entirely at the expense of consumption, which will undergo a sharp contraction. In Figure 10, this adjustment is represented by the jump moving the economy from the stable manifold $V_{NB}^S$ to $V_B^S$ in correspondence to $dk_t = dk^{TH}$. From that moment on, the economy will begin to move toward the steady state through small further reductions both in capital and consumption.

Up to this point, we have fully characterized both the decentralized as well as the planned economy. It is therefore legitimate to ask which are the relationships between the allocations emerging in the two models and whether, in particular, it is possible to establish some Pareto efficiency criteria for evaluating the market economy performance. This in particular raises the question of whether it is possible to decentralize the allocations chosen by the social planner using the tools offered by the fiscal policy, namely the public healthcare supply $\bar{n}$ and the debt-to-capital ratio $\hat{b}$. Recall that, in fact, by continuously adjusting the discount factor $\rho$ from zero to one, we obtain a continuum of Pareto-efficient solutions, characterized by increasing levels of capital, consumption, and healthcare spending. However, it should finally be noted that the parameter $\beta$ measuring the disutility of waiting time in the social planner problem does not appear in the decentralized economy. This therefore requires, as we will do in the next section, to calibrate it appropriately if we would reproduce the optimal solutions resulting from the market economy.

# 4 Decentralizing Pareto Optima

In this Section we carry out the analysis of the decentralization through the market mechanism of the optimal intertemporal allocation of resources chosen by the benevolent social planner. As we have seen, the set of Pareto optimal equilibria is indexed by the social planner's discount factor $\rho$ capturing her degree of "patience". Recall that optimal allocations also include the dynamic sequence of public healthcare $n_t$ (which represents only an imperfect substitute for private healthcare $l_t$) entailing not only a use of resources but also a cost in terms of disutility. However, since the social planner does not face any problem of adjusting the supply and demand for public healthcare, she assumes the coefficient $\beta$ measuring the disutility associated with public healthcare consumption as constant and therefore not time-dependent.

Up to this point in the paper, we have set this $\beta$ parameter arbitrarily, without advancing any hypotheses regarding its relationship with the other economy's structural parameters. We now aim to demonstrate that, if we want to establish some functional equivalence between the planned
economy and the decentralized one, and if, in particular, we want to enable the stationary solutions of the planned economy to be decentralized through an appropriate calibration of public debt-to-capital ratio $\hat{b}$, then the choice of $\beta$ is no longer arbitrary. This is quite understandable if we observe that $\beta$ appears in the equations that fix the stationary values of the variables emerging in the planned economy, while it does not appear in the decentralized one. Furthermore, the stationary capital in the planned model does not depend on $\beta$, but only on the discount factor $\rho$, beside on technology.

Given these considerations, we can then ask whether it is possible to appropriately calibrate $\beta$ so that the stationary values found in the decentralized economy, corresponding to capital, consumption, private and public healthcare, and investment in preventive medicine, coincide with their respective values chosen by the social planner. In other words, we now undertake a reconstruction effort consisting of comparing the stationary expressions of the endogenous variables and determining which restrictions must be imposed on the structural parameters so that the values obtained in the decentralized and centralized economies coincide. Obviously, this task must be performed both when constraint (51) on private healthcare binds and when it does not.

Assume first that (51) does not bind. We can now add the suffix DC to the stationary variables obtained in the decentralized economy and, as before, the suffix SP to those obtained in the social planner problem. Proceeding in this manner, we have proved that equation (62) provides the stationary capital $k^{SP}$ chosen by the social planner. We now ask how it is possible to decentralize such solution as well as all the other ones associated to it. To carry out such a proposal, we first conjecture that, through an appropriate choice of the debt-to-capital ratio $\hat{b}$, it is possible to decentralize $k^{SP}$, i.e., to set $\hat{b}$ such that $k^{DC} = k^{SP}$. Later, of course, we will need to prove the consistency of this conjecture. Note that setting $\hat{b}$ in order to have $k^{DC} = k^{SP}$, one gets that the stationary consumption $c^{DC}$ in the decentralized economy corresponding to a non-binding constraint (62) is

$$c^{DC} = \frac{\alpha}{n^{DC}\beta^{DC}} \qquad (67)$$

while the stationary consumption $c^{SP}$ in the planned economy is

$$c^{SP} = \frac{1 - \bar{\delta}}{n^{SP}\beta^{SP}}.$$

Setting $n^{DC} = n^{SP} = \bar{n}$, where $n^{SP}$ stands for the stationary public health care in the planned economy and $n^{DC} = \bar{n}$ the government supply of public healthcare in the market economy, it is immediate to verify that the requirement to have $c^{DC} = c^{SP}$ is to calibrate the disutility parameter $\beta^{SP}$ in the following way

$$\beta^{SP} = (1 - \bar{\delta})\beta^{DC} = \frac{(1 - \bar{\delta})(1 + \alpha)}{n^{SP}(\gamma(k^{SP})\gamma(1 + \hat{b}) + n^{SP})} \qquad (68)$$

It is also easily verifiable that, under calibration (68), stationary private healthcare $l^{SP}$ in the planned economy and stationary private healthcare $l^{DC}$ in the market economy coincide. In fact, in view of (56), (55), and (67) one has $l^{SP} = \alpha c^{SP} - n^{SP}$ and $l^{DC} = \frac{\alpha}{\bar{n}\beta^{DC}} - n^{SP}$. Equality $l^{SP} = l^{DC}$ thus requires $\alpha c^{SP} = \frac{\alpha}{n^{SP}\beta^{DC}}$ which under calibration (68) boils down to

$$\frac{1 - \bar{\delta}}{n^{SP}\beta^{SP}} = \frac{\alpha}{n^{SP}\beta^{DC}} \qquad (69)$$

and under (68) is satisfied. Finally, we demonstrate that under calibration (68), stationary investment in preventive medicine is the same in the decentralized and planned economies, i.e.
$I^{SP} = I^{DC}$. In view of (68) and (69), one in fact has

$$I^{SP} = \alpha c^{SP} \rho = \alpha \frac{1 - \bar{\delta}}{n^{SP} \beta^{SP} \rho} = \alpha \rho \frac{\gamma(k^{SP})^{\gamma}(1 + \hat{b}) + n^{SP}}{(1 + \alpha)} \quad (70)$$

and

$$I^{DC} = \alpha \frac{\gamma(k^{DC})^{\gamma}(1 + \hat{b}) + \bar{n}}{\gamma(k^{DC})^{\gamma-1}(1 + \alpha)}. \quad (71)$$

However, since $\gamma(k^{SP})^{\gamma-1} = 1/\rho$, one obtains

$$I^{SP} = \rho \alpha \frac{\gamma(k^{SP})^{\gamma}(1 + \hat{b}) + n^{SP}}{(1 + \alpha)} = I^{DC}.$$

The analysis carried out up to this point has been based on a comparison of the planned economy with the decentralized one, starting from the hypothesis that in both economies constraint (51) on private health was not binding. As we have seen in the previous sections, this requires the level of capital, and therefore GDP, to be large enough, namely larger than $k^{TH}$ defined in (64). Since, through the calibration (68) of $\beta$, we have proved that all the variables evaluated in the steady state in the planned and decentralized economies coincide, the critical capital stock $k^{TH}$ will separate the domain of definition of the non-binding regime from that of the binding one in both the planned and decentralized economies. However, from the expression (64), we observe that the capital threshold $k^{TH}$ is a decreasing function of the debt-to-capital ratio $\hat{b}$. As a consequence, it is possible to express the function $\hat{b}_{TH}(k)$ relying $\hat{b}$ to $k$ such that the combinations $(k, \hat{b})$ ensure capital to stick on its critical value $k^{TH}$, as

$$\hat{b}_{TH}(k) = \frac{\bar{n}k^{-\gamma}}{\alpha\gamma} - 1. \quad (72)$$

As it is immediate to verify, (72) is monotonically decreasing from $+\infty$ when $k \to 0$ to some finite constant when $k \to 1$. We can then move on the study of the decentralization of the Pareto optimum in the case where the non-negativity constraint (72) on private health is binding, i.e. when $k < k^{TH}$. From equations (57) and (58), we have the following stationary conditions within the planned economy and involving consumption, public health and investment in preventive medicine

$$c^{SP} = \frac{\bar{\delta} n^{SP}}{\alpha - \beta^{SP} (n^{SP})^2} \quad (73)$$

$$I^{SP} = \alpha \frac{\bar{\delta} n^{SP}}{\alpha - \beta^{SP} (n^{SP})^2}. \quad (74)$$

Note that the corresponding stationary values in the decentralized economy are given by

$$c^{DC} = \frac{n^{DC}}{\alpha}, \quad (75)$$

$$\beta^{DC} = \frac{\alpha}{(n^{DC})^2} \quad (76)$$

and

$$I^{DC} = \frac{\alpha c^{DC}}{\rho \gamma k^{\gamma-1}} = \alpha c^{DC} = n^{SP}.$$
Setting again $n^{DC} = n^{SP}$ and $\beta^{SP}$ according to (68), one can write

$$c^{SP} = \frac{\bar{\delta}n^{SP}}{\alpha - (1 - \bar{\delta})\beta^{DC}(n^{SP})^2} = \frac{\bar{\delta}n^{SP}}{\alpha - (1 - \bar{\delta})\frac{\alpha}{(n^{SP})^2}(n^{SP})^2} = \frac{n^{DC}}{\alpha} = c^{DC}$$

and

$$I^{SP} = \alpha \frac{\bar{\delta} n^{SP}}{\alpha - (1 - \bar{\delta}) \beta^{DC} (n^{SP})^2} = \alpha \frac{\bar{\delta} n^{SP}}{\alpha - (1 - \bar{\delta}) \frac{\alpha}{(n^{SP})^2} (n^{SP})^2} = n^{SP} = I^{DC}.$$

We have thus shown that by setting a stationary level of capital belonging to the set of Pareto optima (which are indexed with respect to the time discount factor $\rho$) and assuming that this level of capital can be decentralized, it is possible to decentralize also all the other endogenous variables, by means of the calibration (68) for $\beta$. We must now demonstrate that our conjecture is consistent, meaning that it is possible to decentralize a given level of capital chosen by the social planner. We now prove that any stationary capital chosen by the social planner can be decentralized by means of an appropriate calibration of the debt-to-capital ratio $\hat{b} \in (-1, \frac{1-\gamma}{\gamma})$. To this end, assuming a non-binding constraint (51), i.e. $k > k^{TH}$, let us take equation (27) evaluated at the steady state and solve it for $\hat{b}$ to obtain

$$\hat{b}_{NB}(k) = \frac{(1 - \gamma)k^{\gamma} - \frac{\bar{n}}{\gamma} \frac{\alpha}{1+\alpha} k^{1-\gamma} - \left(1 + \frac{\alpha}{1+\alpha}\right) k - \bar{\delta}\bar{n}}{\frac{\alpha}{1+\alpha} k + \gamma k^{\gamma}}. \quad (77)$$

On the other hand, when constraint (51) binds, let us solve for $\hat{b}$ equation (34) to get

$$\hat{b}_{B}(k) = \frac{(1 - \gamma)k^{\gamma} - (1 + \alpha)k - \bar{n}\bar{\delta}}{\alpha k + \gamma k^{\gamma}}. \quad (78)$$

In the next Proposition we characterize the functions $\hat{b}_{NB}(k)$ and $\hat{b}_{B}(k)$ and, in particular, we show that both give rise to bell-shaped curves.

**Proposition 13.** The functions $\hat{b}_{NB}(k)$ and $\hat{b}_{B}(k)$ are continuous and differentiable in $k \in (0,1)$. Moreover, $\lim_{k \to 0} \hat{b}_{NB}(k) = \lim_{k \to 0} \hat{b}_{B}(k) = -\infty$, $\lim_{k \to 1} \hat{b}_{NB}(k) = -\left(1 + \frac{\bar{n}}{\gamma} \frac{\alpha}{1+\alpha} + \bar{\delta}\bar{n}\right) < \lim_{k \to 1} \hat{b}_{B}(k) = -\left(1 + \frac{\bar{n}\bar{\delta}}{\alpha+\gamma}\right) < 0$, and both $\hat{b}_{NB}(k)$ and $\hat{b}_{B}(k)$ have two zeros and possess an unique maximum. As a consequence, $\hat{b}_{NB}(k)$ and $\hat{b}_{B}(k)$ generate bell-shaped curves intersecting each other once or twice in $(0,1)$. The first intersection occurs in correspondence to a point lying on $\hat{b}_{TH}(k)$.

*Proof.* Continuity and differentiability are trivial to prove. By checking (77) and (78) one immediately verifies that $\lim_{k \to 0} \hat{b}_{NB}(k) = \lim_{k \to 0} \hat{b}_{B}(k) = -\infty$, $\lim_{k \to 1} \hat{b}_{NB}(k) = -\left(1 + \frac{\bar{n}}{\gamma} \frac{\alpha}{1+\alpha} + \bar{\delta}\bar{n}\right)$, $\lim_{k \to 1} \hat{b}_{B}(k) = -\left(1 + \frac{\bar{n}\bar{\delta}}{\alpha+\gamma}\right)$. The existence of two zeros for both $\hat{b}_{NB}(k)$ and $\hat{b}_{B}(k)$ comes directly from Propositions 3 and 6 applied to the case $\hat{b} = 0$. To prove that $\hat{b}_{NB}(k)$ possesses an unique maximum, we compute its derivative and set it equal to zero

$$\frac{\partial \hat{b}_{NB}(k)}{\partial k} = \frac{\left[ \gamma(1-\gamma)k^{\gamma-1} - \frac{\bar{n}}{\gamma} \frac{\alpha}{1+\alpha} (1-\gamma)k^{-\gamma} - \left(1+\frac{\alpha}{1+\alpha}\right) \right] \left( \gamma k^{\gamma} + \frac{\alpha}{1+\alpha} k \right)}{\left( \gamma K^{\gamma} + \frac{\alpha}{1+\alpha} K \right)^2} - \frac{\left[ (1-\gamma)k^{\gamma} - \frac{\bar{n}}{\gamma} \frac{\alpha}{1+\alpha} k^{1-\gamma} - \left(1+\frac{\alpha}{1+\alpha}\right) k - \bar{\delta}\bar{n} \right] \left( \gamma^2 k^{\gamma-1} + \frac{\alpha}{1+\alpha} \right)}{\left( \gamma k^{\gamma} + \frac{\alpha}{1+\alpha} k \right)^2} = 0.$$
After some straightforward although tedious computations, one obtains that the zero for the derivative of $\hat{b}_{NB}(k)$ are those $k$ solving

$$k^{\gamma} \left[ \gamma(1-\gamma) \left(1 + \frac{\alpha}{1+\alpha}\right) + (1-\gamma)^2 \frac{\alpha}{1+\alpha} \right] - \bar{n}\bar{\delta}\gamma^2 k^{\gamma-1} + \bar{n}\frac{\alpha}{1+\alpha}(1-2\gamma) = k^{1-\gamma} \left[ \bar{n} \left( \frac{\alpha}{1+\alpha} \right)^2 \right] + \bar{n}\frac{\alpha}{1+\alpha}\bar{\delta}. \quad (79)$$

It is easy to see that when $k \to 0^+$ the LHS of (79) goes to $-\infty$ and the RHS is converging to $\bar{n}\frac{\alpha}{1+\alpha}\bar{\delta}$. Moreover, in (0, 1), LHS grows faster than the RHS, implying the existence of a crossing point $\hat{k}$ in (0, 1) between the LHS and the RHS.

Analogously, to verify the existence of an unique maximum for $\hat{b}_B(k)$, we set to zero its derivative. After some straightforward computations, one obtains

$$\frac{\partial \hat{b}_B(k)}{\partial k} = \frac{[(1-\gamma)\gamma k^{\gamma-1} - (1+\alpha)](\alpha k + \gamma k^{\gamma}) - [(1-\gamma)k^{\gamma} - (1+\alpha)k - \bar{n}\bar{\delta}](\alpha + \gamma^2 k^{\gamma-1})}{(\alpha k + \gamma k^{\gamma})^2} = 0.$$

By rearranging terms, one has that the derivative of $\hat{b}_B(k)$ vanishes for any $k$ solving

$$\alpha \bar{n} \bar{\delta} + \bar{n} \bar{\delta} \gamma^2 k^{\gamma-1} = k^{\gamma} (1-\gamma) (\gamma+\alpha). \quad (80)$$

Nevertheless, it is immediate to verify that the LHS of (80) is decreasing while its RHS is increasing. As a consequence, in view of the boundary conditions, one immediately proves the existence of an unique $k$ such the derivative of $\hat{b}_B(k)$ vanishes. Finally, to prove that $\hat{b}_{NB}(k)$ and $\hat{b}_B(k)$ intersect each other once or twice in (0, 1), observe that $\hat{b}_B(k) < \hat{b}_{NB}(k)$ if and only if

$$(1 - \bar{\delta}) \frac{\alpha}{1 + \alpha} \bar{n} + (1 + \bar{n} \bar{\delta}) \alpha + \frac{\alpha^2}{1 + \alpha} \frac{\bar{n}}{\gamma} k^{1-\gamma} - \frac{\alpha}{1 + \alpha} k < \frac{\alpha^2}{1 + \alpha} k^{-\gamma}. \quad (81)$$

It is easy to verify that the LHS is monotonically decreasing in (0, 1) while the RHS give raise to a bell-shaped curve starting from some positive constant when $k = 0$. As a consequence, there will be at most two intersections between the two curves. Before the first intersection it is $\hat{b}_B(k) > \hat{b}_{NB}(k)$ while between the first and the second intersection the opposite holds. Eventually, from the second intersection on, it is again $\hat{b}_B(k) > \hat{b}_{NB}(k)$. Finally, we prove that $\hat{b}_{NB}(k)$ and $\hat{b}_B(k)$ intersect in correspondence to a point $(k, \hat{b})$ satisfying (72). This requires $\hat{b}_{NB}(k) = \hat{b}_{TH}(k)$ and $\hat{b}_B(k) = \hat{b}_{TH}(k)$, i.e.

$$\frac{(1 - \gamma)k^{\gamma} - (1 + \alpha)k - \bar{n}\bar{\delta}}{\alpha k + \gamma k^{\gamma}} = \frac{\bar{n}k^{-\gamma}}{\alpha\gamma} - 1$$

and

$$\frac{(1 - \gamma)k^{\gamma} - \frac{\bar{n}}{\gamma} \frac{\alpha}{1+\alpha} k^{1-\gamma} - \left(1 + \frac{\alpha}{1+\alpha}\right) k - \bar{\delta}\bar{n}}{\frac{\alpha}{1+\alpha} k + \gamma k^{\gamma}} = \frac{\bar{n}k^{-\gamma}}{\alpha\gamma} - 1.$$

Adding one to both sides of the equations above, factoring their left-hand sides and simplifying, we can rewrite them as, respectively

$$\frac{k^{\gamma} - k - \bar{n}\bar{\delta}}{\alpha k + \gamma k^{\gamma}} = \frac{\bar{n}k^{-\gamma}}{\alpha\gamma}$$

and

$$\frac{k^{\gamma} - k - \bar{\delta}\bar{n} - \frac{\bar{n}}{\gamma} \frac{\alpha}{1+\alpha} k^{1-\gamma}}{\frac{\alpha}{1+\alpha} k + \gamma k^{\gamma}} = \frac{\bar{n}k^{-\gamma}}{\alpha\gamma}.$$
Multiplying the last two equations by their respective denominators and rearranging them, we get

$$k^{\gamma} - k - \bar{n}\bar{\delta} = \frac{\bar{n}k^{-\gamma}}{\alpha\gamma} (\alpha k + \gamma k^{\gamma})$$

and

$$k^{\gamma} - k - \bar{\delta}\bar{n} = \frac{\bar{n}k^{-\gamma}}{\alpha\gamma} \left( \frac{\alpha}{1+\alpha}k + \gamma k^{\gamma} \right) + \frac{\bar{n}}{\gamma}\frac{\alpha}{1+\alpha}k^{1-\gamma}.$$

We can thus manipulate the two above equations in order to write

$$k^{\gamma} - k - \bar{n}\bar{\delta} = \frac{\bar{n}k^{-\gamma}}{\alpha\gamma} (\alpha k + \gamma k^{\gamma}) = \frac{\bar{n}k^{1-\gamma}}{\gamma} + \frac{\bar{n}}{\alpha}$$

and

$$k^{\gamma} - k - \bar{\delta}\bar{n} = \frac{\bar{n}k^{1-\gamma}}{\gamma(1+\alpha)} + \frac{\bar{n}}{\alpha} + \frac{\bar{n}}{\gamma}\frac{\alpha}{1+\alpha}k^{1-\gamma} = \frac{\bar{n}k^{1-\gamma}}{\gamma} + \frac{\bar{n}}{\alpha}.$$

The intersections of $\hat{b}_{NB}(k)$ with $\hat{b}_B(k)$ thus occur at a point lying on $\hat{b}_{TH}(k)$. Finally, the intersection between $\hat{b}_{NB}(k)$, $\hat{b}_B(k)$ and $\hat{b}_{TH}(k)$ must occur in the increasing branches of $\hat{b}_{NB}(k)$, $\hat{b}_B(k)$ at a point $\tilde{k}$ such that for any $k < \tilde{k}$ one has $\hat{b}_B(k) > \hat{b}_{NB}(k)$. □

After having fully characterized the functions $\hat{b}_{NB}(k)$ and $\hat{b}_B(k)$, the next Proposition ensures that any optimal allocation can be decentralized in the market economy by setting the debt-to-capital ratio $\hat{b}$ according to (77) or (78).

**Proposition 14.** Provided $\beta^{SP} = (1 - \bar{\delta})\beta^{DC}$, any social planner solution can be decentralized by fixing opportunistly $\hat{b}$. More specifically, for $\rho$ such that the pair $(k, \hat{b})$ lies below the curve (68), $\hat{b}$ must be fixed according to (78), while for $\rho$ such that the pair $(k, \hat{b})$ lies above the curve (68), $\hat{b}$ must be fixed according to (77).

Proposition 14 thus explains how to decentralize any optimum through the choice of $\hat{b}$. Indeed, when $k < k^{TH}$, $\hat{b}$ must be chosen on the ground of $\hat{b}_B(k)$ while, when $k > k^{TH}$, on the ground of $\hat{b}_{NB}(k)$.

In Figure 11 we have depicted the graphs of the functions $\hat{b}_{NB}(k)$ and $\hat{b}_B(k)$ (respectively, solid blue and red lines) and we have underlined in black the composition of the two functions consistent with the binding and non-biding regimes (dotted black line). We thus get that the decentralization rule corresponds to the superior envelope of the functions $\hat{b}_{NB}(k)$ and $\hat{b}_B(k)$. In the previous Proposition, we have seen that it is always possible to decentralize any allocation chosen by the social planner by appropriately calibrating the debt-to-capital ratio. Furthermore, for low levels of capital, as one wishes to decentralize economies characterized by larger levels of capital, it is necessary to increase the debt-to-capital ratio. Subsequently, when economies begin to be endowed with larger capital stocks, their decentralization requires decreasing debt-to-capital ratios. This non-monotonic, bell-shaped trend in the optimal debt-to-capital ratio, seemingly inconsistent with standard results in the literature, can be easily understood. To this end, note that an increase in public debt crowds out productive investment in agents' savings, thus slowing down growth and leading to lower steady-state capital levels.

However, in the market economy one observes the coexistence of two stationary solutions, ranked with respect to their capital intensity. When debt-to-capital ratio increases, the lower steady state increases while the higher one decreases. As a consequence, if the decentralization procedure is entrusted to the low steady state of the market economy, it will be necessary to
<figure><img src="image_14.png" /><figcaption>Figure 11: Optimal debt-to-capital ratio</figcaption></figure>

increase the debt-to-capital ratio to attain a larger target. By contrast, when one wishes to decentralize high levels of capital through the highest steady state of the decentralized economy, then it will be necessary to reduce the debt-to-capital ratio to enhance the process of capital accumulation.

# 5 Concluding Remarks

In this paper, we have studied the global equilibrium dynamics of an *OLG* economy in which agents derive utility directly from their health status. From this perspective, alongside ordinary consumption, they invest in preventive medicine in their youth, while in old age they turn to both the private and public health systems to meet their health needs. The market economy is described by the composition of two dynamic regimes depending on whether the level of capital is low or high and the resulting demand for private healthcare is zero or positive. In any case, there emerge two stationary solutions; the low one is globally unstable, the high one being globally stable. An innovative result consists in obtaining a countercyclical dynamic of waiting times, since as income increases, the demand for healthcare shifts toward the private sector, easing the pressure on public healthcare and thus reducing average waiting times. Faced to the typical dynamic efficiencies of *OLG* models, we proceeded to identify the set of Pareto-optimal allocations, which are indexed with respect to the social factor of time discounting. Finally, we have shaped the fiscal policy in such a way it allows efficiency of the market economy to be restored.

Several extensions for further research naturally emerge from our analysis. A first one could consist in introducing a mechanism of intergenerational transmission of health capital. In our model, indeed, health status of each generation depends on individual choices in terms of preventive medicine. However, in the real world, health endowment of a newborn cohort is strongly correlated with the health status of its parents, through both biological and socio-economic channels. Embedding this transmission into the *OLG* structure would allow us to analyze the long-run propagation mechanisms of health status. At the same time, it would
highlight the cumulative impact of policy interventions across successive generations.

A second line of research would involve the explicit modeling of life expectancy. In the present framework, survival in old age is assumed to be certain, while in practice longevity depends on health investments, environmental conditions, and public health policy. Allowing life expectancy to be endogenous would make it possible to capture the two-way link between health and demography. In an *OLG* economy, changes in longevity directly affect the sustainability of pension and healthcare systems, the distribution of fiscal burdens across generations, and the dynamic efficiency of capital accumulation. Incorporating such a channel would thus provide a powerful tool to address the macroeconomic challenges of the aging of the population.

Another extension concerns the role of uncertainty and exogenous shocks. Health expenditures, technological innovations in medicine, and demographic dynamics are all subject to unexpected changes. Introducing stochastic shocks into the model would allow us to study how the system reacts to demographic or epidemiological crises, such as pandemics, as well as to sudden improvements in medical technologies.

Finally, one could enrich the model by considering the interaction between health and the environment. This is widely discussed in the “one health” literature, as environmental quality is a major determinant of humanity health status. In this way, it would be possible to analyze joint policies in health and environmental domains. Public policy in fact often faces the dual challenge of financing healthcare while also tackling environmental degradation. Introducing environmental externalities alongside health dynamics would allow for a more comprehensive analysis of sustainable welfare and intergenerational equity.

While our model already offers a useful framework to study a large number of issues, the extensions outlined above would greatly broaden its scope and enhance its overall relevance. We thus leave such proposal for future research.

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