A new waste and energy nexus? Rethinking the modernisation of waste services in Delhi

One important aspect of sustainability involves the flows of materials and energy, extracted, consumed, transformed and disposed of in the functioning of urban societies, which makes it directly linked to the 'nested' character of infrastructure that this special issue of Urban Studies on 'urban nexus' is keen to address. In particular, the question of urban waste, a sector previously neglected in the field of urban policy, has slowly become a major issue in world urbanisation that can be tackled through its cross-sectorial interactions and its multidimensional effects. Through an analysis of the case of Delhi, this article aims to undertake an exploration of the waste and energy nexus in order to contribute to the current debates on the socio-technical transformations of waste infrastructure and its societal interlinkages. This article studies the effects of waste management policies in Delhi that essentially promote large centralised technical systems such as waste-to-energy plants, which are presented as a 'modern nexus' of waste and energy, at the expense of any 'alternative nexus' such as the existing traditional recycling sector. Hence, the main objective is not only to question the socio-spatial and political implications of the current reforms of the waste sector but also to discuss the development of other potential decentralised solutions that could complement the overall system in an adapted way.

Introduction

The theme of sustainable urban development at the Habitat III Conference reconfirms the key role cities play in the regulation of major global challenges. One important aspect of sustainability involves the flows of materials and energy, extracted, consumed, transformed and disposed of in the functioning of urban societies, which makes it directly linked to the 'nested' character of infrastructure that this special issue of Urban Studies on 'urban nexus' is keen to address (as posited by Jochen Monstadt and Olivier Coutard in their proposal for this special issue). In particular, the question of urban waste, a sector previously neglected in the field of urban policy, has slowly become a major issue in world urbanisation that can be tackled through its cross-sectorial interactions and its multidimensional effects ( UN-Habitat, 2010). The organised treatment of this waste has become a strategic tool in urban marketing that serves to consolidate the attractiveness of the most exemplary cities in this field, such as the 'zero waste' policies of Adelaide, San Francisco and Stockholm (Zaman and Lehmann, 2013), and obliges the others to align with this ideal of modernity. In India, while the rapid urbanisation of the last two decades has done much to improve living standards, it has also been accompanied by significant environmental deterioration, with at least 125,000 tons of solid waste generated daily by cities (CPCB, 2012). These phenomenal quantities were long neglected by the authorities, before becoming a key target of public action from the mid-1990s and the launch of environmental policies. This willingness has been strongly reaffirmed under the framework of the national '2014-2019 Swachh Bharat Abhiyan' (Clean India Mission), which aims to improve the collection, treatment, disposal and reuse of waste flows. Through an analysis of the case of Delhi, the country's capital and showcase, this article aims to undertake an exploration of the waste and energy nexus in order to contribute to the current debates on the socio-technical transformations of waste infrastructure and its societal interlinkages. Home to 16 million people (in Delhi National Capital Territory) and part of an agglomeration of over 25 million inhabitants, the city generates around 9000 tons of domestic waste (according to municipality official records, January 2017) to be managed daily. Complementary to similar surveys in other contexts (see, for example, in Germany, Florentin, 2016), this article studies the effects of waste management policies in Delhi that essentially promote large centralised technical systems, such as waste-to-energy plants,1 which are presented as a 'modern nexus' of waste and energy, at the expense of any 'alternative nexus' such as the existing traditional chain of the recycling sector (widely documented since the early 1980s; see, for example, the pioneering work of Furedy (1984) or Mukherjee and Singh (1981)). In Delhi, 150,000 to 200,000 informal waste workers (Gidwani and Chaturvedi, 2011b: 131) recover recyclable materials in parallel to the formal municipal waste system, and transfer it to informal recycling units. Most of them belong to marginalised groups, particularly Dalits2 or deprived Muslim communities. This work constitutes an essential livelihood for them (Gill, 2012), while providing an environmental service to the city. Thus, the main objective of this article is not to give a technological analysis of the system's ongoing reforms, but rather to resituate this issue in cities characterised by severe social inequalities. More specifically, it aims more to resituate the issue at the level of the sociotechnical implications related to the ambiguities of a process of infrastructure modernisation. In particular, a solution like incineration that is claimed to follow the principle of resource efficiency seems controversial as it is based on materials with high calorific potential, specifically plastics, paper and cardboard, all products traditionally recycled by informal retrievers (see photographic illustrations by De Bercegol and Gowda, 2016). Hence, the focus is not only to question the socio-spatial and political implications of the current reforms of the waste sector but also to discuss the development of other potential decentralised solutions that could complement the overall system. The data presented here was collected during several visits to India between 2013 and 2017, in the context of the 'Syracuse' project,3 followed by another French research project 'Orva2d'4 financed by the French Development Agency,5 dealing specifically with the issue of waste recovery. During this time, we conducted approximately 40 semi-directive interviews with actors involved in the formal sector: engineers, politicians and private agencies who showed us the treatment sites. A large part of the fieldwork also included more ethnographic observations of meetings and follow-up with so-called 'informal' actors who are part of the waste recycling chain, from collection to treatment plants: collectors, retrievers, resellers and recyclers. Finally, around 20 members of civil society were also interviewed, at the level of resident associations involved in non-governmental organisations and militant associations. This article presents some of the results of these surveys, with a critical analysis of the waste management choices made in Delhi and a plea for a better articulation of the existing resources. To begin with, relying on existing scientific literature on the subject, in the following section we provide a brief background of the socio-spatial ambiguities of the waste modernisation policies in Delhi, which have dramatic social consequences for the precarious labour conditions of waste workers. This state of affairs brings us to our research question on how to rethink the waste-to-energy nexus in terms of situated resource efficiency linked to the informal recycling economy. Although allegedly qualified as 'non-modern' practices by the public authorities, waste sorting and the recycling sector appear to be a legitimate energy-efficient solution. Therefore, we provide a socio-technical analysis of the current reorganisation of the waste sector in Delhi, highlighting not only the deficiencies of the current centralised system but also the potentialities of integrating decentralised solutions for improved management. Finally, we conclude by giving some recommendations by conciliating the existing different nexus towards more sustainable waste management practices in developing cities like Delhi. The urban politics of waste sector modernisation and the making of the city Eviction of 'non-modern' practices As in other developing cities around the world, the modernisation of waste management in Delhi can be seen as a reflection of a 'sanitary ideal' (Melosi, 2000) that seeks to replicate the technical solutions developed in the North, within which the informality of poor neighbourhoods, and their 'non-modern' waste recycling practices, have little or no place. With regard to the similar process in Cairo, Bénédicte Florin notes that it is the 'ideology' of modernisation (Florin and Cirelli, 2015) that makes public decision makers consider informal sector actors as 'ineligible' for integration into the new planned system. The urban elite and creditors (for whom 'the collection systems in India are primitive' (World Bank, 2008: 19)) see recycling workers as backward and incompatible with 'modern' waste management (Wilson et al., 2006), which necessarily has to be placed in the hands of private organised companies and to employ mechanised equipment on the pretext of sustainable development. For many slum dwellers whose survival depends on waste recovery,6 the exclusion of their everyday experience is then reinforced by a 'dispossession' of their means of subsistence (Schindler et al., 2012). The privatisation of the sector, and the choice to incinerate waste on the pretext of solving the waste crisis and generating energy out of it, are thus not innocuous socio-technical choices. They reflect a social choice that embodies the primacy of capitalist accumulation over the power of work: the process [of waste incineration] can be seen as an example of accumulation by dispossession, which means there is an inherent necessity of the owners of capital to separate, by force if necessary, labourers (waste workers) from the means of production (waste, transfer stations, etc). Once this process is complete, a small number of waste workers are offered the opportunity to sell their labour for a wage in waste-to-energy plants. (Schindler et al., 2012: 20) When describing the informal recyclers of electronic waste in Bangalore, who are experiencing a similar situation of dispossession, Reddy proposes the concept of 'abjection', which describes not only the injustice of being excluded, but also the humiliation of being expelled from a sector to which they have always belonged (Reddy, 2015: 171). Reddy utilises the metaphor of waste to describe 'the humiliating experience of informal recyclers becoming -to invoke Wright (2006) -figures of waste' (Reddy, 2015: 171). Thus the promotion of a so-called 'modern' management system, based on Western models, reveals the desire to increase control over the economy by relegating the informal actors involved. Relegation of the poor population At a wider level, planning policies tend to deny the existence of populations confined to informality and produce 'urban exclusion' which pushes these populations away to the periphery (Kundu and Sarangi, 2005). The political economic works on the rise of Indian civil society and its gentrification (Ghertner, 2011) explain how the theme of environmental protection has made it possible to legitimise the relegation of poor populations, on the pretext of their adding to urban pollution. Since 1990-2000, reforms encouraging civil participation7 have directly contributed to consolidating the growing influence of the powerful Resident Welfare Associations (RWAs) in planned neighbourhoods. Paradoxically, this has led to marginalisation of the poorest categories, by successfully imposing the interests of a sociologically dominant class in municipal councils to the detriment of a poor and less well organised population (Zérah, 2007). The poor population's lack of representation is particularly evident in the judicial sphere, as legal mechanisms require financial and temporal means, which are not amongst these people's prerogatives. Thus, mobilised by the middle classes, the judicial institutions (the Supreme Court of India, the High Court of Delhi and the Green Tribunal) have become important arbitrators whose decisions significantly impact urban governance, especially in conflicts concerning the environment, like those involving waste, promoting ecological considerations to the detriment of their social consequences. This results, for example, in the plastic recycling workshops in the north of the city being shut down regularly after a ruling by the Green Tribunal (30 units closed at Nangloi and Mundka in June 2015, where the emblematic PVC market, where waste plastic is bought, separated and resold in large volumes, has been relocated after being progressively relegated from the city centre to the periphery of Delhi; see Gill, 2012). Baviskar's research on 'bourgeois environmentalism' allows us to understand how environmental discourses pronounced by the middle class in Delhi insidiously contribute to driving the poor out of the city: 'for the bourgeois environmentalist, the ugliness of production must be removed from the city […] urban spaces should be reserved for white-collar production and commerce, and consumption activities' (Baviskar, 2002: 41). Legal decisions are made for local environmental reasons, with no consideration for more global environmental considerations (a reduction in the use of raw materials, for example) or for those in insecure employment. The bourgeois vision of the environment, reinforced by legal decisions in its favour, takes precedence over social considerations and amplifies the insecure situation of poor urban populations (Mawdsley, 2004). We can hence understand that the modernisation of the waste sector is part of the 'urban renewal' policies implemented in Delhi which, in reality, are a euphemism for the demolition of insalubrious areas. Rehabilitation in the periphery, on less expensive land, is symbolic of an evacuation towards the outer fringes, which sometimes worsens the inhabitants' living conditions (Dupont and Vaquier, 2014). The 'modernisation' of infrastructures thus clearly falls within a process of exclusion that relegates the poor to deteriorated peripheries, outside cities. Vinay Gidwani and Rajyashree N. Reddy thus put forward the concept of 'eviscerating urbanism' (Gidwani andReddy, 2011: 1640), which is particularly relevant in light of the last Delhi Master Plan and its 'vision' of a 'global city and a world-class metropolis' that has got rid of its pollution and its slums (DDA, 2010: 2). Ghertner (2011) underscores the cultural importance of this policy of aestheticisation in the construction of the contemporary city where the environmental ideal is based on the ideal model of the 'world class city'. The city is created by the middle class, to replicate Paris, New York or Singapore-reference cities, within which the poor have no place. This is actually a process of remodelling urban space to conform to the aesthetic models of the upper classes. Rethinking the nexus between waste and energy Following the recent academic work on waste transformations in Delhi, the infrastructural transformations taking place in the waste sector in Delhi can be read as a pretext of pollution used to evict the town's proletarian categories (Demaria and Schindler, 2016;Gidwani and Reddy, 2011;Gill, 2012), through the introduction of hegemonic environmental polices of neoliberal inspiration which reveal the 'violence of sustainable urbanity' (Swyngedouw, 2014). But, this does not directly explain why the current reorganisation of the technical system appears to be unsuccessful at solving the waste crisis despite massive investment to modernise its infrastructure. Why do the chosen technical options to facilitate a nexus between waste and energy appear to be a 'false good idea' in the context of Delhi? By choosing to align with the 'world class city' reference, the mimicry of the urban planning authorities in Delhi tends to eliminate the option of the emergence of a site-specific model, freed from the commands of Western modernity, while improving its efficiency as well as local conditions. In this article, we assume that this replication might not be the best option to fit the particularities of the waste sector in Delhi, and our research enquires about the potentialities of situated alternative solutions that could be more resource efficient and socially inclusive. Therefore, while the critical debate on the subject is generally approached in terms of 'deficiencies', we would like to take into account the forms of creativity the actors demonstrate by identifying potential co-production regimes specific to Delhi. What are the local potentialities for an improved way of managing waste in an Indian city? We make the hypothesis that there is a nexus between waste and energy, where the 'people [of the informal recycling sector act] as infrastructure' (in Simone's (2004) sense), which appears to be highly efficient despite its pitfalls, these latter partly due to its public relegation to informality. To underscore the relational effects between the current modernisation and this informal nexus, this article borrows particularly from urban political ecology as, by evacuating the waste flows generated by urban society, waste management directly contributes to the urban metabolism, to be understood here as all the processes by which agglomerations utilise, consume and transform resources. Since the end of the 20th century, the linear economy for the sanitary disposal of urban waste has been challenged by a circular economy paradigm valuing the recovery of waste resources. From the perspective of a circular metabolism, 'informal' channels for the retrieval of waste, which have been in existence for a long time, contribute to the reduction of environmental pressure and appear to be in that sense an efficient nexus between waste and energy. In the wake of the philosophy behind the 'Syracuse' research project, this article seeks to complete these metabolic analyses by enriching them with socio-spatial features interlinked to infrastructural technical systems that structure the discharge of these flows. Since the 'traditional' recovery practices have not been able to tackle all the increased volumes of waste associated with the current rapid urbanisation, we need to question their place and the role they could effectively play together with other systems. Relying on contemporaneous French academic debates, our research questions the scale of these infrastructural choices and considers their social and political effects. How does the modernisation of the waste sector affect the making of the city in Delhi? Following on from works inspired by science and technology studies (STS), and specifically by research undertaken at Laboratoire Techniques, Territoires et Sociétés8 (Coutard and Rutherford, 2015;Dupuy, 1991), the analysis focuses on the transformation of infrastructure because 'the choices made at this level end up structuring the daily existence [of urban populations]' (Lorrain, 2011: 32, translated from French). Technical public services, in this case the waste management service, are thus highly structuring 'instruments' of public action, instilled with a political meaning that reflects the transformation of relationships between the State and society (Lascoumes and Le Galès, 2007). By looking at the transformation of the waste sector in Delhi, the aim is thus to interpret its political meaning in order to ultimately question the making of the city. Socio-technical analysis of the ongoing transformations: Local constraints and specific opportunities of Delhi's waste systems Vast, ineffective, modern systems For the administrators, the problem of solid waste management in Delhi is first and foremost a problem of land, given that the growing volumes (around 9000 tons daily according to municipal official records in 2017) demand ever more space in a city where it is in short supply and put the available treatment facilities under intense pressure (see Table 1). Since 1975, a succession of 23 landfill sites9 has been created. Seventeen have been closed after reaching their maximum storage capacity, and two others for legal reasons. In 2014, therefore, four landfill sites were still running, three of which had already exceeded their maximum capacity and lifespan (10 to 20 years) but were still being used by the municipal authorities despite being officially declared illegal by the Delhi Pollution Control Committee. These are the Okhla site in South Delhi (1996, 16 hectares), Ghazipur in the East (1984, 29 hectares) and Bhalswa in the North (1994, 21 hectares). Although the opening of a new 40 ha site in Narela-Bawana in 2012 took some of the load off North-East Delhi, the Ministry of Urban Development estimates that 600 ha will actually be needed by 2021 to handle the waste properly and to be able to close the old landfill sites. However, available land is scarce in Delhi, and resident groups oppose any new project in the vicinity of their neighbourhoods. The Delhi Development Authority (DDA) is finding it hard to identify a new site: 'there is simply no more land available to accommodate the need of proper sanitary landfills in Delhi' (according to a retired DDA official, January 2016). In the meantime, the waste is piling up (the Okhla mountain is more than 180 feet high) with no real controls (no gas extraction or leachate recovery, except in the most recent facility), which creates legal conflicts between municipalities, government and citizens. In these conditions of land scarcity, incineration technologies look like a life-saving solution for the authorities. In 1989, the Ministry of New and Renewable Energy, in cooperation with the government of Denmark, ran a first incinerator experiment with the target of incinerating 300 tons per day and generating 3.7 MW of electricity, but the site closed after just three weeks because the humidity levels of the incoming waste were too high and their calorific value too low (550-850 kcal per kg, whereas the minimum required for successful operation should be 1200-1400 kcal per kg). After this first failed attempt, the government recently commissioned three private operators to manage three new waste-to-energy plants. However, the opening of the Ghazipur incinerator (1300 tons per day, 10 MW), operated by ILFS,10 and of Narela-Bawana (4000 tons per day, 24 MW), operated by Ramky,11 has been postponed several times in the four years until March 2017 (see Table 2). This delay was due to disputes over the rate for selling power to the major distribution companies (which was set at 2.49 rupees/kWh against a needed 6 rupees/kWh to ensure the financial viability of the process, according to a Jindal manager interviewed in January 2016), as well as technical reasons because of the low calorific value of the incoming waste. This low quality of waste explains why waste-to-energy plants operate below their capacity, at an average of 1000-1500 tons per day (according to the monthly data provided by the municipality between January 2016 and January 2017) instead of the 2500 tons initially planned in their design inception. According to data submitted by the operator of the Okhla incinerator (2000 tons per day, 16 MW), run by Jindal,12 which has been operational since 2009, between 2011 and 2015 the incoming waste at the gate of the incinerator was composed on average of 69% biodegradable matter (food and garden waste), 30% high calorific value waste (wood, plastic, paper, rubber, textiles) and 1% inert waste (see Table 3). In order to run properly, these incinerators need a continuous supply of waste with a high calorific value (such as paper and plastic), and this puts them in direct competition with the informal waste pickers, estimated to number 150,000 in Delhi (according to Chintan13 official representatives interviewed in January 2016), who pick up the best paper and plastic at their sources, in order to resell in a recycling chain that is ignored by the authorities and the engineers. As less than a third of the incoming waste is of high calorific value, the operator is obliged to mechanically dry the remaining waste in order to be able to burn it (as is the case at the Jindal plant), or even to transform it to combustible fuel pellets (as at the ILFS plant at Ghazipur), which in both cases consumes energy and is a costly process. These technical problems are further exacerbated by legal opposition, with the emergence of coalitions of opponents demanding a halt to these incineration technologies, such as the ongoing court case against the Jindal plant (Sukhdev Vihar Residents' Welfare Association & Others v. State of NCT of Delhi & Others). The coalitions are motivated by various philosophies, ranging from the 'Not in My Backyard' grievances of resident associations located next to the plants, to NGOs representing the ragpickers' needs or environmental justice claims (on these unexpected coalitions, see Demaria and Schindler, 2016). They accuse the plants of being polluting and dangerous, although paradoxically the Jindal plant is recorded with the United Nations Framework Convention on Climate Change (UNFCCC) and is eligible to receive carbon credits for turning waste into electricity (see http://cdm.unfccc.int/Projects/DB/SGS-UKL1185291186.52). For their detractors, 'this plant should not be granted any environmental benefit: it is unscientific, anti-environment and anti-public health' (according to the head of the Toxics Watch Alliance,14 January 2016). Waste incinerators seem likely to further exacerbate an already catastrophic atmospheric situation: concentrations of particulates (less than 2.5 microns) are almost 15 times higher than WHO norms, caused not only by monstrous traffic jams but also by power stations located in the city (which are mostly based on coal), and above all by the dust exuded from the innumerable construction sites typical of a fast-growing city. Similarly, centralised composting appears to be quite inefficient. The first composting plant was established in the 1980s at Okhla with a planned capacity of 150 tons per day. It was halted between 1991 and 1995, before being forced to restart by a ruling of the Indian Supreme Court, and then closed again and replaced by the incinerator. Three other units were established, in Bhalswa (1998, 500 tons per day, closed in 2015) and in Tikri (2001, 125 tons per day, closed in 2013), but only the one in Okhla (1985, capacity 500 tons per day) is currently operating (see Table 4). The opportunities for selling compost (10-20 rupees/kg) are limited by the subsidised prices of chemical fertiliser (5.5 rupees/kg). Furthermore, built in the middle of the city, centralised compost plants in Delhi require transportation to link the production to the distant agricultural fields. Because the waste is coming in unsegregated, the non-biodegradable waste has to be separated at the plant itself by manual and mechanical means, increasing production costs, and the compost produced remains of low quality due to a lack of homogeneity, as it is difficult to remove all the plastic and glass. For example, although the Okhla unit, run by ILFS since 2007, can boast having received the equivalent of €40,000 in carbon credits in 2013 (without which it would have made a loss, according to the site manager interviewed in December 2014), it is only running at maximum half of its capacity of 500 tons per day and only produces some 30 tons of mediocre quality compost, which is difficult to sell, despite being bought by private fertiliser companies (for less than €20 per ton). In both cases, centralised composting as well as incineration, capturing the waste at the source to allow appropriate segregation and thus control waste quality, is one of the most basic ways to limit the constraints of these technologies. Nevertheless, for the moment, the municipal collection system still begins officially at the dhalaos, intermediate neighbourhood-scale collection points with a capacity of 12 to 16 tons designed to cover residential groupings of 10,000 to 15,000 residents, from which waste is transferred to the centralised treatment centres (compost plan, incineration, landfill). Since 2005, virtually all collection from these transfer stations has been assigned to large, private, indigenous operators (Ramky, ILFS, as well as SPML).15 These private operators are paid according to the quantity collected ('more waste, more money'). In theory, since 2000, the legislation on waste (Municipal Solid Waste Rules 2000) requires municipalities to collect waste no longer from the dhalaos, but directly from households, with the aim of improving the operation of the processing treatment systems (whose efficiency depends on the quality of the incoming waste). However, in 2015, only 10-15% of the city centre was actually receiving this service, from the firm Ramky. In 2016, the updated legislation (Municipal Solid Waste Rules 2016) and awareness raising campaigns among the population restated the need for door-to-door collection, in order to provide better fuel for the centralised treatment of waste-to-energy. Nevertheless, door-to-door collection has yet to come into practice. Industrialised collection with trucks coming to pick up waste at the household level is technically difficult to achieve given the narrow streets in Indian cities (according to municipal officers) and because it has to compete with a complex system of waste pickers who informally collect the most valuable materials at source (according to a SPML manager interviewed in December 2014). Nevertheless, investors are betting that they will slowly succeed in privatising collection at the household level in order to control the entire chain, from the source to the centralised treatment plants. A traditional waste sector in a state of radical upheaval The reorganisation of the waste sector is politically sensitive in Delhi as its industrialisation is in direct competition with an already existing but informal collection of reusable waste recovery, the basis of the informal recycling economy. The public authorities are reluctant to recognise these unofficial but existing collection practices, as they raise appropriation conflicts with private formal firms (Cavé, 2014). These informal waste pickers cover the city, recovering useful waste that can be sold to wholesalers before being transferred to specialist markets where the materials -sorted, cleaned and processed in recycling workshops -are sold at the end of the chain for three to four times their initial value (on interlinked contracts, patronage and exploitation in the waste recovery market in India, see Gill, 2007). It has been estimated that the total social value added from waste selling activities in Delhi could be around 3587 million rupees (US$56 million) per year (Hayami et al., 2006: 63-64). While calls for tenders for an organised door-to-door collection were issued in December 2014 to the large private organised sector in order to extend this collection to the city as a whole, advocacy groups working on behalf of the waste pickers, such as Chintan in Delhi, or unions like All India Kabari Mazdoor Mahasangh, Safai Sena,16 are fighting for the informal sector to be recognised and fairly included: 'Collection has to be formally given to those who have been doing it since forever and are already helping to reduce the waste crisis' (according to a Chintan manager, January 2016). Chintan estimates that by informally collecting waste directly from residents, the informal collectors save the municipality 15 million rupees (US$307,000) per day, based on their being paid the (2003) minimum wage like the 60,000 municipal sweepers are. Recognising the existence of local recycling initiatives would improve the efficiency of the waste management system and contribute to reducing the mass of waste. Despite competition with formal private operators, the sector continues to recover and to treat vast quantities of materials: paper, plastics, glass and metals (Cavé, 2014;Gidwani and Chaturvedi, 2011b;Gidwani and Reddy, 2011). Nevertheless, it remains extremely difficult to assess given its informal nature. Our empirical data provides a rough assessment based on a sample of 15 families or groups of workers in Hanuman Mandir, a slum in South Delhi that is mainly inhabited by ragpickers who collect waste from the neighbouring middle class areas. We followed these groups for two weeks in December 2016 in Safdarjung Enclave, an upper middle class area next to the slum. We found that an average of 40-50 kg of waste materials (excluding glass bottles) per family were sold daily to a local wholesaler. For a municipality of Delhi's scale, this could mean that possibly approximately 2000-2500 tons of waste per day will be informally diverted in order to be recycled, based on a conservative estimate of 50,000 waste picking families working in Delhi (Chintan, January 2016). This would be equivalent to 20% of the waste generated daily in Delhi, which agrees with the figure commonly advanced by NGOs and ragpickers' advocates. Despite its informality, the sector is very well organised in India as it relies on dynamic cast networks from the collection of materials to their separation and transformation. For example, Gill has shown how the Khatik community (a lower cast from Rajasthan) has specialised in trading plastic scraps in Delhi by developing for themselves the skills to be able to differentiate all types of plastics (polypropylene, polyethylene, polyvinyl chloride, polyurethane, etc.). Waste pickers ultimately appear more skilled and efficient than the formal system in terms of gathering and segregating waste (Gidwani and Chaturvedi, 2011a: 59), which has also been shown by research in other developing countries (Gunsilius et al., 2011). For Chintan, it is therefore not the waste-to-energy plants which should be UNFCCCregistered but the groups of workers, as they contribute to reducing the amount of waste sent to landfill sites, avoiding greenhouse gas emissions of 900,000 tons per year (Chintan, 2012). These waste picker advocates would like to see the introduction of a management model inspired by the SWaCH (Solid Waste Collection and Handling) cooperative in Pune,17 a big city in the State of Maharastra. Advocated as one of a kind in India, this model consists of a partnership between a union of 6000 workers, the Kagad Kach Patra Kashtakari Panchayat, and the municipality, which has led to the regularisation of 2300 workers in part of the city. To a certain extent, this formalisation has been reproduced in certain areas of Delhi by resident welfare associations. One example in place since 2003 is Defence Colony, one of the upmarket wards of South Delhi, where the resident welfare association for Block-A (around 1000 households), in cooperation with the NGO Toxics Link,18 has set up a decentralised management system that has led to the regularisation of 17 workers. These workers were paid 100 rupees each per dwelling per month in 2016 for the in situ sorting of waste. Six pits were dug to receive part of the green waste and to produce good quality compost, which sells easily to local residents and provides a source of income to finance the maintenance of the system (protective clothing for the workers, purchase of carts, digging of new pits). Although it is still necessary to reject half of the waste supplied to to the dhalaos, this decentralisation of the system allows it 'to reduce significantly the amount of waste to be evacuated to the centralised treatment facilities by 50%, [and has] notably improved the social conditions of the workers' (Defence Colony Resident Welfare Association's representative, January 2016). Interestingly, the organised private sector is trying to take over this model, which remains for the moment organised by residents' associations only and is not yet integrated into the municipal collection service. After the privatisation of the dhalaos, the organised private sector has now planned to contractualise workers, for example with the arrival of ILFS to undertake door-to-door private collection in the blocks of South Delhi under its new subsidiary Dakshin Dilli Swachh Initiatives Limited (DDSIL -South Delhi Clean Initiatives Limited),19 in order to access waste with high calorific value before it gets evacuated towards informal recycling. One can see here the implicit recognition by the private sector of the positive dynamics of the informal traditional actors in segregating waste at its source. In particular, one small start-up company called Pompom,20 created by an employee of Ramky, follows directly the same dynamics by innovating with a mobile phone application to facilitate the sale of waste by its users to the company, which comes directly to homes to collect it and eventually resells it to the informal chain of recycling industries: 'don't throw it, give it to me, waste is a valuable resource' (according to a Pompom manager, December 2014). This modern replication of the traditional Kabadiwala waste collection system is in itself a proof of its legitimacy and efficiency, which clearly demonstrates the essential role that a collection at a decentralised scale has to play for the recovery of waste. Similarly, another neighbourhood that exemplifies the craze for 'zero waste' in Delhi is New Moti Bag, a neighbourhood of around 1100 households essentially inhabited by the families of senior civil servants, where a private company has taken over the waste management. The company Green Planet Waste Management, financed by the plastics sector within the framework of an eco-responsible approach,21 has invested around €150,000 since 2014 in a decentralised treatment technology that would theoretically enable all waste to be reprocessed in situ: 'Evacuating waste to a landfill is a complete heresy; we have the capacity to treat most of it at the source and reduce the burden to the environment', said the director of the company in January 2016. The biodegradable waste is indeed treated on site, to create pellets and compost to be sold, along with plastic bottles, using the operator's own technology. All the remaining dry waste is sorted on site and sold to the informal recovery sector (according to a Green Planet Waste Management manager, January 2016). Here also, this innovative solution is not a panacea in itself as the manager complained about the lack of financial outcomes of the system because he was expecting support via the allocation of advertising hoardings for rent, which was not finally given by the government. Finally, in a city characterised by very high levels of inequality, waste retrieval is one of the main means of subsistence for a large number of poor households, and their sole employment: 'we make a living of waste' said a ragpicker (Hanuman Mandir, January 2016). If the sector was better integrated, the workers' jobs would be guaranteed, from the lowest level collectors, to the industrialists at the top of the pyramid, along with the profusion of professions specialised in the collection, sorting, resale, cleaning, crushing and transformation of materials. This waste recovery economy provides a basic income for hundreds of families and ultimately, at the end of the chain, sustains formal industries that produce consumer goods. Regulation of the recycling sector and its inclusion in the waste management process would enable the progressive introduction of social and environmental norms in order to avoid the numerous socio-environmental pitfalls of the sector. There is hence a need to go beyond the sordid image of a sector that is doubtless imperfect in many respects but where it is in fact the condition of informality that tends to facilitate abuses, especially in terms of exploitation and child labour. Without recognition of the legitimacy of the sector and proper public support, the precarious labour conditions of the workers will keep deteriorating. This can already be seen in the implementation of the Okhla waste-to-energy plant for the ragpickers living in its vicinity. Here, workers' earnings have decreased by 21% according to a survey based on a sample of 109 persons working there (Chintan, 2012: 12). As stated by the Alliance of Indian Wastepickers: informal workers are highly vulnerable, needing special and urgent attention. Strong measures like increased social security need to be undertaken to reduce the risk. In the coming years, we must commit ourselves to the campaign of social security for informal workers and pursue 'Right to Social Security' to become a justiciable right. In addition to that, questions of housing, education, skill-building, nutritious food, clean environment all need answers preferably sooner than later. (Arora, 2016) All in all, the question of a 'right to the city' arises (Lefebvre, 1968), understood here in its reformist sense, as a series of rights in the city (Zérah et al., 2011), or in other words, the recognition of fundamental rights, including the right to a livelihood (theoretically guaranteed by the Indian Constitution).22 The current reform of the waste sector ignores this right to the city for waste workers, and neglects an existing nexus, more inclusive and already energy efficient. Conclusion: Ways towards an inclusive and efficient waste nexus Rethinking the nexus between waste and energy is a fruitful method to analyse critically the ongoing reform agenda of the waste sector in Delhi and to understand better its effects on the making of this city. Waste recovery in Delhi seems to be central to the modernisation of the sector, but the authorities' choice of a modern nexus technology seems to be incomplete, if not flawed, in terms of adapting to the social complexity of this huge metropolis. The Delhi case illustrates some of the factors hindering the introduction of decentralised solutions in the big cities of the South which could lead to more appropriate nested socio-technical systems of waste management. The waste crisis in Delhi is seen above all as a technical problem to be tackled as such: by technicians, and through a technical infrastructure based on a centralised system. After the failure of over-exploited landfills, the limitations of this technical approach are visible in the recent incineration systems. Although the latter reduce real estate pressure in terms of the final storage space, their performance for the time being remains mediocre from an energy, environmental and economic viewpoint, particularly given the mismatch between the technology and the composition of the waste produced in Delhi. Nonetheless, the investments made by major operators are likely to slowly lead to an improvement in incineration techniques, but at a very high cost, while energy recovery from waste should ensure the economic viability of an increasingly expensive activity. However, as stated in the projections of the planning commission itself, there is so much waste in Delhi that even if it were possible to incinerate it and compost it at the maximum efficiency by 2021, there would still remain a minimum of 4000 to 5000 tons that would have to be put in landfills (DDA, 2010: Chap. 10). Furthermore, while it can certainly improve the efficiency of the industrialised treatment, particularly by capturing waste at source, in the short term, the ongoing liberalisation of the waste sector remains highly conflictual, politically sensitive and devoid of convincing techno-economical results.

The decentralised private or community-based alternatives favoured by the opposition for their part provide a potential for waste reduction that could possibly efficiently support the vast technical systems that they cannot realistically replace given the challenge of the colossal volumes of waste to be handled. In the current situation, it seems difficult to imagine the replication on a metropolitan scale of decentralised management experiments developed at the neighbourhood level. In a metropolis with a population of over 16 million inhabitants, there are numerous restrictions on their development: socio-spatial limitation (these experiments emerge in wealthy, middle class residential areas, while Delhi suffers greatly from urban poverty); land restrictions (they require the allocation of space that is difficult to obtain in regulatory and legal terms, particularly in densely populated zones) and technico-economic restrictions (they underpin the regulation in clusters of different industrial channels of recovery and the need to reorganise commercial outlets, particularly for compost, which must be sold for financial gain). By taking into account the city's social, land and institutional fragmentation, a solution could lie in the regulation of an appropriate coordination of the informal recycling sector and the vast, centralised, technical incineration systems, in order to create a more inclusive and efficient nexus. A collaboration between the formal and informal sectors (Chaturvedi et al., 2015), with a minimisation at the source of the waste generated (Narain and Singh Sambyal, 2016), appears to be the key to solving Delhi's waste management crisis. To achieve this, one could imagine a progressive hybridisation of scales of intervention, with a semidecentralised type of system: -by encouraging household sorting to facilitate a reduction at source, particularly by promoting the production of compost at the neighbourhood level and its local reuse in gardens and streets; -by formalising a door-to-door collection system by local workers who have the expertise to segregate the best recyclable materials, in order to improve the efficiency of recycling; -by connecting the disposal of the residual waste, mainly biodegradable, with a municipal system better calibrated in terms of size and process, such as methanisation technology which is yet to be developed in India. By locally reducing the quantity of waste, this reorganisation of the system could make it possible to reduce the size of the land surfaces required for landfills. A recognition of the importance of the informal recycling sector in any event seems to be fundamental to improving its efficiency. The improvement of capturing and sorting conditions could further increase the volumes of recycled materials, reduce the overall cost of waste treatment for everyone and support the urban economy by providing stable employment to a large number of workers. A public campaign that associates the skills of the subaltern town, by recognising the economics of recycling, with those of the residents' daily lives, by encouraging sorting at source, could be a first major step towards facilitating the sustainable management of Delhi's waste and moving towards an inclusive urban nexus. The new 2016 waste management regulations that, albeit implicitly, recognise the existence of the informal sector seem to be opening the doors to a new collaborative phase. Their introduction could provide an opportunity to catalyse a more sustainable and inclusive waste management system. Nevertheless, formalising the traditional chain of recycling is not in itself a panacea to solve the waste crisis. One has to closely analyse the conditions in which this option can be articulated with a wider reform framework. Given this perspective, we suggest that further research should be conducted to understand the modalities of (non-)cooperation between the different stakeholders involved in order to facilitate more efficient and inclusive waste management in developing cities and to imagine the development of a new waste and energy nexus.

Table 1 .

 1 The landfills of Delhi (on 4 January 2016).

	Site Date of opening	Size Number of trucks on 4 January 2016 Tons received on 4
	January 2016 Last expected closing date
	Ghazipur	1984 29ha 508 2104 2008
	(East Delhi)
	Bhalswa	1994 21ha 385 2965 2005
	(North Delhi)
	Okhla 1996 16ha 221 1142 2006
	(South Delhi)
	Narela-Bawana (North Delhi) 2012 40ha 169 1687 n.d.
	Source: Official municipal data for 4 January 2016.

Table 2 .

 2 Waste-to-energy plants in Delhi.

	Location of waste-to-energy plants Company	Capacity of electricity generated
	Theoretical capacity of waste processing	Amount of waste processed on 4
	January 2016
	Okhla (South Delhi) Jindal 16 MW 2000 tons per day	1165 tons
	Ghazipur (East Delhi) ILFS 10 MW 1300 tons per day	Opening delayed to end of 2016
	Narela-Bawana (North Delhi) Ramky 24 MW 3000 tons per day	Opening delayed to
	March 2017
	Source: Official municipal data for 4 January 2016.

Table 3 .

 3 Characterisation of incoming waste at the Jindal waste-to-energy plant.

	Waste type Monitoring period: 30 March 2011 -31 August 2012	Monitoring
	period:
	1 September 2012 -31 December 2012	Monitoring period: 1 January 2013 -30 June
	2015 Average 2011-2015
	Metric tons %	Metric tons %	Metric tons %	%
	Biodegradable					69%
	Food waste 74,732 35% 565,471	42% 46,575 30% 40%
	Garden waste 95,857 45% 365,202	27% 35,949 24% 29%
	High calorific value						30%
	Paper/cardboard	16,831 8%	137,649	10% 17,565 12% 10%
	Textiles5438 3%	93,335 7%	9533 6%	6%
	Wood 5179 2%	122,636	9%	29,437 19% 9%
	Rubber and leather 2846 1%	13,389 1%	7432 5%	2%
	Plastics 3794 2%	44,011 4%	6529 4%	3%
	Other, inert waste	7871 4%	0	0%	0	0	1%
	Source: Author's compilation of Monitoring Reports 2012 and 2015 (monitoring period
	numbers 1 and 2) for the Timarpur-Okhla Waste Management Company Pvt Ltd's integrated
	waste-to-energy project in Delhi, submitted to UNFCCC for CDM:
	http://cdm.unfccc.int/Projects/DB/SGS-UKL1185291186.52 (accessed 10 April 2018).

Table 4 .

 4 Compost plants in Delhi.

	Location of compost plants Company	Theoretical capacity of waste processing
	Amount of waste processed on 4 January 2016
	Okhla ILFS 500 tons per day	161 tons
	Bhalswa	Nature and Waste Management Pvt Ltd	500 tons per day	Closed
	since 2015
	Tikri ILFS 125 tons per day	Closed since 2013
	Source: Official municipal data for 4 January 2016.

Acknowledgments

The authors are thankful to the guest editors and reviewers for their valuable comments, as well as to Aliya Tuzhilin for her editing work on this text.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research would have not been possible without the financial support of the French National Research Agency (http://www.agencenationale-recherche.fr/en/about-anr/about-the-french-national-research-agency/), under the project 'Syracuse' (http://www.agence-nationale-recherche.fr/Project-ANR-11-VILD-0008), and the French Development Agency (https://www.afd.fr/en), under the project 'Orva2d' (http://eso-lemans.cnrs.fr/fr/recherche/programmes-en-cours/projet-afd.html). Footnotes 1. Waste-to-energy is a form of energy recovery which consists of generating electricity (or heat) through combustion of waste. GO TO FOOTNOTE 2. Dalit is a term used to describe the group of people previously regarded as 'untouchable' who are traditionally very often relegated to carrying out tasks regarded as impure by upper castes in India. GO TO FOOTNOTE 3. This programme was a research funded by the French National Research Foundation. It aimed 'at examining the most promising technological innovations in the field of urban utilities, mainly around issues of symbiosis, convergence and decentralization, and at attempting to create an instrument of evaluation of their degree of pertinence according to the context' (http://www.agence-nationale-recherche.fr/?Project=ANR-11-VILD-0008). GO TO FOOTNOTE 4. Orva2d: http://eso-lemans.cnrs.fr/fr/recherche/programmes-en-cours/projet-afd.html. GO TO FOOTNOTE 5. Agence Française de Développement: http://www.afd.fr/lang/en/home. GO TO FOOTNOTE 6. The figures on this, which are inevitably controversial, differ from one institution to another, ranging between 11% (according to the official Census of India of 2011) and one third of slum dwellers living in Delhi (Banda and Sheikh, 2014). GO TO FOOTNOTE 7. Two important dates can be mentioned: in 1996, a complaint commission was created in Delhi to hear the grievances of discontented citizens. This facilitated the emergence of legal militancy and the condemnation of public authorities for negligence in dealing with waste management and, at a wider level, urban pollution. In 2005, the Right to Information Act (RTI) provided citizens with better transparency regarding administrative actions, particularly with regard to municipal expenditure and contracts signed with the private sector. GO TO FOOTNOTE 8. Laboratoire Techniques, Territoires et Sociétés: https://latts.fr/.

GO TO FOOTNOTE 9. The term 'landfill site' corresponds here to the official denomination used by the municipal authorities, implicitly referring here to the technical appellation of 'sanitary landfill'. In practice this is a simple dumping site without proper sanitary infrastructure to manage gas emanations, which are nevertheless minimally controlled thanks to a protective wall surrounding the site. GO


Laurent Denis
email: laurent.denis@univ-lemans.fr

Xavier Fairise
email: xavier.fairise@univ-lemans.fr

Ze Zhong

A Life-Cycle Model with health spending and endogenous life time

Keywords: Health stock, health spending, lifespan, life-cycle

, we suppose that the depreciation rate of health stock increases with age. Lifespan is thus an endogenous variable. We also suppose that the multiplicative productivity term of the income function (depending on the health of the agent) displays a hump shaped profile over time. Our contribution is to investigate the Grossman (1972)'s model in a non stationary environment with an endogenous finite lifetime. After derivation of the optimality conditions the model is solved numerically and we simulate the path followed by the main variables over the agent's life-cycle. Calibrated on the US economy with realistic values of parameters, the model is able to replicate a growing profile of health spending and a significant share of health spending at the end of life roughly in accordance with the data.

Introduction

Health spending holds a growing place in economic debate. At the macroeconomic level, health spending represents a growing share of the GDP. At the microeconomic level, health spending is a key variable in agent choices over the life-cycle. Moreover, it tends to represent a growing share in agents' spending over their life cycle. The issue of health spending and its growing trend may stem from several factors: ageing, scientific innovations or new medical treatments, etc.. Ageing leads to an increase in demand and scientific innovations lead to an increase in supply (new treatments, quality improvement,...).

In this article, we are interested in studying health spending decisions over the lifecycle. What are the main features of health spending over the life-cycle ? French, Mc-Cauley, Aragon, et al. (2017) provide a lot of data concerning end-of-life medical spending in a large set of countries. They focuse particularly on the last three years (and the last year) of life. In most countries, medical spending made at the last year of life represents 8 to 11% of the aggregate medical spending (8.5% in the US). Concerning the last three years of life, the medical spending made during this period represents between 16.7 and 24.5% of the aggregate medical spending made over the life-cycle(16.7% in the US). Even if we consider the heterogeneity between countries, it is clear that medical spending in the last three years of life represents a significant share of the aggregate medical spending. Using US data, [START_REF] Halliday | Health Investment over the Life-Cycle[END_REF] and [START_REF] Fonseca | Retirement and the Demand for Health[END_REF] report an increasing profile of medical spending over the life-cycle, medical spending being approximatively zero around 20-25 years old. [START_REF] Jung | Medical Consumption over the Life-Cycle: Fact from U.S. Medical Expenditures Panel Survey[END_REF] reach similar conclusions. They also estimate the evolution of the share of health spending. At 20-25 years old, health spending is zero while at 85, it roughly represents more than 50% of income. The profile is increasing and convex. Finally, we underline that at the aggregate level, medical spending represent around 17% of the GDP in the US.

From a theoretical point of view, [START_REF] Grossman | On the Concept of Health Capital and the Demand for Health[END_REF] develops a framework including a modeling of health spending. [START_REF] Grossman | On the Concept of Health Capital and the Demand for Health[END_REF]'s model is a life-cycle one and he considers the health level may be represented by a stock variable, that is the health stock or health capital. The health stock has a positive impact on the agent's welfare. It depreciates at a rate possibly increasing with age. The health stock can also be improved or maintained by continuously health investments. Finally, the income of the agent is an increasing function of his health. If the health stock becomes less than a threshold, the agent dies, lifespan is an endogenous variable. The health capital concept is in the continuity of the human capital concept introduced among others by [START_REF] Ben-Porath | The Production of Human Capital and the Life Cycle of Earnings[END_REF]. The agent has also access to a financial market on which a risk-free asset is exchanged. [START_REF] Grossman | On the Concept of Health Capital and the Demand for Health[END_REF]'s model includes several other features we will not detail. [START_REF] Grossman | On the Concept of Health Capital and the Demand for Health[END_REF]'s model is at the base of a vast literature. [START_REF] Halliday | Health Investment over the Life-Cycle[END_REF] and [START_REF] Fonseca | Retirement and the Demand for Health[END_REF] use life-cycle models including an health stock variable, an endogenous health spending and a financial asset. They evaluate the ability of their models to quantitatively replicate some stylized facts. [START_REF] Fonseca | Retirement and the Demand for Health[END_REF] are interested by retirement choices in an uncertain environment. They simulate the path of some key variables and discuss their adequacy with their empirical counterpart. [START_REF] Halliday | Health Investment over the Life-Cycle[END_REF] focus on the growing profile of health spending over the life-cycle. Their model, with endogenous labor supply, is calibrated, numerically solved and simulated. They discuss the ability of their model to replicate the profile of some variables such as consumption or medical spending.

Some authors theoretically study the [START_REF] Grossman | On the Concept of Health Capital and the Demand for Health[END_REF]'s model. In order to obtain analytical results and to isolate the key mechanism, they use a simplified version (often in continuous time) with only a health stock variable and without a financial asset. [START_REF] Laporte | Should the Grossman Model of Investment in Health Capital Retain its Iconic Status[END_REF] derives some results concerning the dynamic of the model (uses of phases diagrams). [START_REF] Strulik | A Closed-Form Solution fir the Health Capital Model[END_REF] obtains a closed-form solution and performs exercices of comparative dynamics. Finally, [START_REF] Laporte | Making the Grossman Model Stochastic: Investment in Health as a Stochastic Control Problem[END_REF] introduce uncertainty into the model.

In the contributions of [START_REF] Laporte | Should the Grossman Model of Investment in Health Capital Retain its Iconic Status[END_REF], [START_REF] Laporte | Making the Grossman Model Stochastic: Investment in Health as a Stochastic Control Problem[END_REF] and [START_REF] Strulik | A Closed-Form Solution fir the Health Capital Model[END_REF], the depreciation rate of health stock is constant over time, which allows the agent to potentially live forever. Indeed, in the Grossman's model, the agent stays alive as long as the health stock is above the threshold. If the health stock depreciation rate is constant, the agent may have the possibility to indefinitely maintain his health stock above the threshold. In this article, we consider a more realistic assumption and suppose that the health stock depreciation rate increases as the agent ages. Lifespan is now an endogenous variable. We also depart from [START_REF] Strulik | A Closed-Form Solution fir the Health Capital Model[END_REF] by supposing that the multiplicative productivity term of the income function (depending of the health stock) is not constant over time. We guess it displays a hump shaped profile over time. This allows to replicate an hump shaped profile of income and to assess its consequences on the agent choices. In order to avoid utility being negative, we use an utility fonction with a strong and positive enough constant, which is in line with that of [START_REF] Rosen | The Value of Changes in Life Expectancy[END_REF] and [START_REF] Hall | The Value of Life and the Rise in Health Spending[END_REF]. This model, written in continuous time, is solved numerically. A realistic calibration with a life span of 85 years shows that our model is able to replicate a growing path of the health spending over the life-cycle and a significant share of health spending especially at the end of life. We also perform a sensitivity analysis, which allows us to evaluate how the numerical results are modified if the value of some key parameters are changed.

The paper is organized as follows. In section 2 we present the model and derive the optimality conditions, in section 3, we discuss the calibration strategy and the numerical results.

A life-cycle model with health capital

Presentation of the model

Our model is based on [START_REF] Grossman | On the Concept of Health Capital and the Demand for Health[END_REF]. We use a simplified version similar to the one developed by [START_REF] Strulik | A Closed-Form Solution fir the Health Capital Model[END_REF], [START_REF] Laporte | Should the Grossman Model of Investment in Health Capital Retain its Iconic Status[END_REF] and [START_REF] Laporte | Making the Grossman Model Stochastic: Investment in Health as a Stochastic Control Problem[END_REF] while keeping the main ingredients of Grossman's model. Agent's health is characterized by its health capital level and agent's utility depends on consumption and health. The income of the agent is an increasing function of the health capital. The agent must allocate its income between consumption and investment in health capital. Finally, we assume that health capital depreciates at a rate increasing with age. The agent ceases to live if the health capital fall below a threshold.

The intertemporal utility function

The representative agent maximizes its intertemporal discounted utility:

∫ T 0 e -ρt ( (C γ t H 1-γ t ) 1-σ 1 -σ + b ) dt (2.1)
with C t the consumption and H t the health capital. T is the agent lifespan which is an endogenous variable. More precisely, we consider t = 0 corresponds to the beginning of the active life (that is 25 years) and T is the lifespan in adulthood (that is T+25 years). ρ > 0 is the discount rate, σ > 1 is the intertemporal elasticity of substitution and γ ∈]0, 1[ is the weight of consumption in utility. It is also possible to interpret C γ t H 1-γ t as an aggregate good produced with private consumption and health capital through a constant return to scales technology. b is a constant positive term. The key role of this parameter is widely discussed by [START_REF] Rosen | The Value of Changes in Life Expectancy[END_REF] and [START_REF] Hall | The Value of Life and the Rise in Health Spending[END_REF]. In particular, the value of b must be sufficiently high to ensure a positive flow of utility. We'll discuss this point in appendix A.

The law of motion of health capital Health capital evolution is described by the following equation:

Ḣt = AI t -δ t H t (2.2)
I t represents the investment in health capital. Health capital is produced through a linear technology, A being the efficiency of this technology. Thus, at each date, the health capital is increased by an amount AI t .

Health capital depreciates at a rate δ t increasing with the age of the agent. Following [START_REF] Fonseca | Retirement and the Demand for Health[END_REF], we guess that δ t could be written as follows:

δ t = δ 1 exp(δ 2 t)
with δ 1 > 0 and δ 2 > 0. δ 1 represents the health depreciation at the beginning of the adulthood.

Death occurs when health capital falls below the threshold H > 0. The agent is still alive at time t if H t ≥ H.

The aggregate resource constraint There is a unique good produced by the household which is consumed and invested. The good is produced by the household through a decreasing return to scales technology. Health capital H t is the unique input. The household's production can also be interpreted as an income in real terms. The agent can improve its health by reducing consumption and increasing investment in health capital.

At each period t, one has:

θ t H α t -I t -C t = 0 (2.3)
where θ t > 0 denotes the efficiency of the production technology. Having assuming a decreasing return to scales technology, one has α ∈]0, 1[. In order to generate an inverted-U profile of income, we will suppose that productivity does the same. One has θ t = θ exp(v 0 + v 1 t + v 2 t 2 ) with θ > and the polynomial parameters v 0 , v 1 and v 2 such that the agent's productivity has an inverted-U profile. Finally, we impose the following positivity condition on investment, I t ≥ 0, ∀t ∈ [0, T ]. This condition is natural since health capital can not be "uninstalled" and sold.

Household's programme

The initial health capital of the agent is H 0 > H. The agent chooses the optimal path of consumption and investment maximizing his lifetime utility subjected to resource constraints. First, let us assume a given lifespan T . The household's program is therefore written as:

max ∫ T 0 e -ρt ( (C γ t H 1-γ t ) 1-σ 1 -σ + b ) dt (2.4) subject to                Ḣt = AI t -δ t H 1 (q t ) θ t H α t -C t -I t = 0 (µ t ) I t ≥ 0 (η t ) H t -H ≥ 0 (ξ t ) H T -H ≥ 0 (2.5)
The optimality conditions The Hamiltonian and Lagrangian associated to the above program ares written as:

H t = (C γ t H 1-γ t ) 1-σ 1 -σ + b + q t (AI t -δ t H t ) L t = H t + µ(θ t H α t -C t -I t ) + ηI t + ξ t (H t -H)
The following optimality conditions are obtained:

∂L t ∂C t = γ (C γ t H 1-γ t ) 1-σ C t -µ t = 0, t ∈ [0, T ] ∂L t ∂I t = q t A -µ t + η t = 0, t ∈ [0, T ] qt = ρq t - ∂L t ∂H t = (ρ + δ t )q t -(1 -γ) (C γ t H 1-γ t ) 1-σ H t -µ t θαH α-1 t -ξ t t ∈ [0, T ]
Otherwise, one has the following exclusion conditions:

η t I t = 0, t ∈ [0, T ], with η t ≥ 0 ξ t (H t -H) = 0, t ∈ [0, T [, with ξ t ≥ 0 q T (H T -H) = 0, with q T ≥ 0
Optimal lifetime We previously considered that the lifetime T was given. In our setting, T must be chosen in such a way it maximizes the agent's intertemporal utility along the optimal path, that is:

∫ T 0 e -ρt ( (C γ t H 1-γ t ) 1-σ 1 -σ + b ) dt
The optimal lifetime T * satisfies the following optimality condition 1 :

H(T * ) = ( (C γ T * H 1-γ T * ) 1-σ 1 -σ + b ) + q T * (AI T * -δ T * H T * )
The agent is alive as long as

H t ≥ H. If at date T , one has H T > H and H T -δ T H T dt ≥ H,
the agent can live an additional time interval dt without investing in health capital. Clearly, T is thus not an optimal life time. We conclude at the optimal time T * , one has

H T * = H.

Numerical investigations

The system of equations can not be solved analytically. The time interval [0, T ] is discretized and the set of equations given the solution of the model is solved numerically using a standard Newton algorithm 2 .

Calibration strategy

The model is calibrated in order to match US data. The discount rate ρ is set to 0.04, which correspond to an annual interest rate of 4%. The risk aversion coefficient take the traditional value of 1.5. As previously discussed, σ must be greater than 1. Parameter α is the elasticity of income with respect to the health stock. We choose α = 0.15, which is of the same order as the value of 0.146 estimated by [START_REF] Fonseca | Retirement and the Demand for Health[END_REF]. Parameter γ (resp. 1γ) corresponds to the weight of private consumption (resp. health stock) in utility. In a static model, this parameter, together with α, determines the share of consumption (and health spending) in income 3 . In USA, this share is roughly of 17% and in France, it is of 11%. Given the retained value of α, a value of γ = 0.95 would be necessary to obtain (in the static model) a health spending 1 This optimality condition giving the optimal time is obtained by deriving the intertemporal utility with respect to T and applying the envelop theorem. This condition can be found in any standard optimal control book.

2 The system of equations to be solved is shown in appendix B.

3 Consider the simplified model:

max C,I,H (C γ H 1-γ ) 1-σ 1 -σ s.t. { θH α -C -I = 0 (λ) AI -H = 0 (µ)
share of 19%. There is no clear evidences in the literature concerning the numerical value of this parameter. For instance, [START_REF] Fonseca | Retirement and the Demand for Health[END_REF] set γ = 1 whereas Halliday, He, Ning, and Zhang (2017) chose γ = 0.64. Parameter γ is set at 0.98, which allows to obtain a health spending share close to 15-16%. The parameters A and θ are normalized to 1.

The profile of θ t over the life cycle determines the income profile. According to the Lee, [START_REF] Lee | Charting the Economic Life Cycle[END_REF], labor income attains a peak around 50 years old. Between 25 and 50 years old, labor income increases by around 70% (see [START_REF] Lee | Charting the Economic Life Cycle[END_REF], figure 5). Using PSID data, Halliday, He, Ning, and Zhang (2017) report a humped shaped profile of labor income over the life cycle. A peak occurs between the age of 45 and the age of 50. Between the age of 25 and the age of 50, labor income roughly increase of 90%. Finally, at age 60, labor income decreased by about 30%. The calibration of the parameters of the productivity function θ t seeks to reproduce the same orders of magnitude concerning the income profile. To determine the values of v 0 , v 1 and v 2 , we proceed as follows. During a period of 25 years (from 0 to t * = 25), we guess that the individual productivity grows from θ 0 to θ * (at age t * ). We impose the normalization θ 0 = θ and we set the following growth rate z 1 = θ * θ 0 with z 1 > 1. We then deduce the values of v 0 , v 1 and v 2 . One has log(θ

t /θ) = v 0 + v 1 t + v 2 t 2 . Knowing that θ 0 = θ, it is immediate that v 0 = 0. Now consider log(θ * /θ) = v 1 t * + v 2 t * 2 . θ t being maximum at time t * , this latter satisfies t * = -v 1 2v 2 .
We deduce the following equation giving the value of

v 2 : log(z 1 ) = v 1 t * + v 2 t * 2 = -v 2 t * 2 .
Setting z 1 = 1.8 and t * = 25, the following values are obtained v 0 =, v 1 = 0.0470 and v 2 = -0.0009 and the resulting income profile is close to its empirical counterpart. Labor income increases by around 70% between the ages of 25 and 50. [START_REF] Fonseca | Retirement and the Demand for Health[END_REF] assume the health stock is between 0 and 100, in other words they choose a 0-100 scale. They also report that the health stock decreases from 73.3 at age 26 to 39.7 at age 85. We calibrate the model in ordre to satisfied the same proportions and we impose H H 0 = 39.7 73.3 . The level of these variables is set in order to have an health investment equal to 0 during the first periods (as it is the case in Halliday, He, Ning, and Zhang (2017) and [START_REF] Fonseca | Retirement and the Demand for Health[END_REF]. One has H 0 = 5.1558 and H = 2.7545. Finally, following [START_REF] Fonseca | Retirement and the Demand for Health[END_REF], the parameters describing the depreciation of health capital take the values δ 1 = 0.035 and δ 2 = 0.025. We impose b = 4, so that the agent lifetime is of around 85 years.

The benchmark calibration is summarized in tables 1 and 2.

Is is easy to determine that:

I C + I = 1 -γ + γα

Numerical results

To begin, we briefly discuss how the benchmark calibration allows to adjust the targeted values. We obtain an optimal T of 60, which corresponds to a lifespan of 85 years. The share of health spending ( I C+I ), that is its average value calculated over the life-cycle, is of 17.17%. Figure 1 presents the evolution of health spending, consumption, health stock and income over the life-cycle. We observe that health spending is zero around from the age of 25 to the age of 35. Income displays a hump shaped profile and it reaches a peak around 50 years. Between 25 and 50 years old, income increases of 70%, next, it decrease of about 8.1% until the age of 60, which is consistent with the data [START_REF] Halliday | Health Investment over the Life-Cycle[END_REF]).

We now discuss how the model fit the data. [START_REF] French | End-of-Life Medical Spending in Last Twelve Months of Life is Lower than Previously Reported[END_REF] report that the medical spending of the last three years represents 16.70% of the US aggregate medical spending. In the other countries they study, this value is around 20%. As we consider a representative agent over his life-cycle, we calculate the share of the last three years health spending in total health spending. We find a value of about 7.14%. This value is less than the one reported by [START_REF] French | End-of-Life Medical Spending in Last Twelve Months of Life is Lower than Previously Reported[END_REF]. However, health spending during the last three years of life have a significant level.

The profile of health expenditures (figure 1) is globally increasing, which is roughly consistent with the data (see [START_REF] Halliday | Health Investment over the Life-Cycle[END_REF] and [START_REF] Fonseca | Retirement and the Demand for Health[END_REF]). However, we underline that the health expenditures profil is concave and during the last years of life, it is slightly decreasing. The decrease of the health stock is not monotonous4 . Health stock displays a slight bump around the age of 60 and it attains its critical threshold H at the age of 85. The bump of the health stock is generated by the hump shaped income profile. The profile of health expenditures is concave, which means that the agent significantly invests in health during the income growth phase. It follows a phase of health capital accumulation from 40 to 60 years old.

Income and consumption displays an hump shaped profile. Consumption grows by 48% between the ages of 25 and 50. It then decreases by 9% between 50 and 60 years old and by 81% between 60 and 85 years old 5 . During the working life, the consumption profile is slightly smoother than that of income. Indeed, in this class of models, agents wish to smooth their consumption over their life-cycle. However, after age 60, consumption decreases dramatically, which does not fit the data.

Our model is able to reproduce an increasing profile of health spending. It also reproduce a hump shaped profile of consumption. We note that consumption decreases very strongly after 60 years old. The hump shaped profile of the productivity θ t generates the hump shape profile of income which in turn determines the profile of consumption. However, the consumption profile is less smooth than in the data. This may be explained by the absence of financial market. The agent cannot transfer income over time and the smoothness of consumption is not enough.

We point out that way to obtain a hump shaped profile of income. Our model looks like life-cycle models with human capital and endogenous wage (see [START_REF] Ben-Porath | The Production of Human Capital and the Life Cycle of Earnings[END_REF]). This class of model is able to generate a hump shaped income profile. In our model, the health stock variable plays a similar role to human capital. We underline two differences. Firstly, we guess the depreciation rate increases with age. Secondly, the elasticity of income with respect to health capital is, in our benchmark calibration, equal to 0.15. This value is too small to generate a hump shaped profile of income. A value of around 0.8 would be necessary to obtain it, which is not an admissible value. Income is less sensitive to health than to education.

Sensitivity analysis

In this subsection, we examine how the numerical results are modified if we change some key parameters.

The elasticity σ We simulated the model for two other value of σ, that is σ ∈ {1.25; 1.5; 2} (Figure 2). Recall that 1 σ is the intertemporal elasticity of substitution. This variable expresses the will of the agent to substitute intertemporally consumption (or more precisely, the "composite good" C γ H 1-γ ). Thus, increasing σ means a reduction of the willingness of the agent to make intertemporal substitutions. In other word, the agent prefers a smoother consumption profile. As we want to evaluate the impact of a perturbation of the elasticity σ on the trade-off made by the agent, the numerator of the utility function 1σ is always parametrized using the benchmark value of 1.5, otherwise, the utility level and the lifespan would be impacted by an other channel that consumption and health stock. In other words, a perturbation of σ would have effects similar to a change in b. Increasing parameter σ leads to a decrease in the lifespan and in the health spending share (table 3). If the elasticity σ is low, the intertemporal elasticity of substitution is high and the agent willingness to smooth consumption is low. As the depreciation rate of health increases with age, the necessary health expenditures are continuously growing. Consequently, the agent should accept a strong and increasing reduction of consumption in order to stay alive. He is willing to do so if the intertemporal elasticity of substitution is low.

Figure 2 presents the paths followed by health spending, consumption, health stock and income for the three values of σ. We can see that the paths followed by consumption and income coincide. We also observe there is now upward shift of consumption as σ increases.

The parameter b

We consider two alternative values for parameter b, that is b ∈ {2; 3.8; 6} (Figure 3). As discussed in A, this parameter plays a key role in the determination of the lifespan. He represents the utility of being alive. The results shown in table 4 point that the lifespan is increasing relative to b. We also observe an increase in the health spending share. This is obviously related to the lifespan increase and the increase in the health stock depreciation rate. Consequently, to live longer, the agent must increase his health spending share as he gets holder. It follows an increase in the heath spending share. 3 presents the paths followed by health spending, consumption, health stock and income for the three values of b. The income and consumption paths are roughly superimposed. In other words, the path of a given value of b approximatively extends the one obtained with a smaller value. An increase in b corresponds to an increase in the utility of being alive. Consequently, the agent is willing to accept a reduction in consumption to finance health spending in order to stay alive. After 55 years old, the paths of health spending and health stock move move upwards as b increases. This reflect the increase of the health spending share.

The parameter γ We suppose that γ ∈ {0.9; 0.98; 1}. Recall this parameter has a direct impact on the behaviour of the agent in regards to health spending.

The more interesting case is γ = 0.9. Under this value of γ, the agent chooses to devote a greater share of his expenses to health spending (Table 5). The health spending is then strictly positive very early (Figure 4). Recall that with higher values of γ, there is no health spending until 35 years old. The paths followed by the health stock and the health spending move upwards as γ decreases. The income profile does the same.

Conclusion

In this article, we study a simplified version of the Grossman (1972)'s model. We mainly follow [START_REF] Laporte | Should the Grossman Model of Investment in Health Capital Retain its Iconic Status[END_REF] and [START_REF] Strulik | A Closed-Form Solution fir the Health Capital Model[END_REF]. The agent can allocate his income between consumption and health spending. The health of the agent is characterized by his health stock or health capital. The depreciation of the health stock may be compensated by health spending. The agent stays alive as long as the health stock is above a threshold. Our main contribution is to investigate the [START_REF] Grossman | On the Concept of Health Capital and the Demand for Health[END_REF]'s model in a non stationary environment with an endogenous finite lifetime. One key assumption concerns the depreciation rate of the health stock. Contrarily to [START_REF] Laporte | Should the Grossman Model of Investment in Health Capital Retain its Iconic Status[END_REF] and [START_REF] Strulik | A Closed-Form Solution fir the Health Capital Model[END_REF], we suppose the depreciation rate increases with age. This last assumption avoids the problem of potential immortality of the agent. We also suppose the productivity term of the income function displays a hump shaped profile over the life cycle. Finally, we point out that there is no financial market.

We derive the optimality conditions that allow to characterize the solution. The optimal control problem presents some difficulties. There are two occasionally binding constraints (health spending is positive or zero and the health stock is greater than or equal to the critical threshold) and lifespan is endogenous.

We characterize numerically the solution of the model. The model is calibrated on the US economy. The model is able to reproduce an increasing profile of health spending over the life-cycle and a significant share of health spending especially at the end of life. The model also reproduces a hump shaped profile of consumption. However, consumption at older ages is too low compared to the data. This may be due to the assumptions of this simplified model. For instance, there is no financial market. The agent can not transfer income through time and cannot smooth its consumption path.

A The constant term in the utility function

Here we discuss the meaning of the term b in the utility function. [START_REF] Rosen | The Value of Changes in Life Expectancy[END_REF] is interested by the question of the value of life and develops theoretical frameworks allowing to highlight its determinants. Two classes of models are considered, models with a survival rate and deterministic life-cycle models. Note that this two classes of models are closely related and have very similar results. In the line of [START_REF] Rosen | The Value of Changes in Life Expectancy[END_REF], we use a deterministic life-cycle model in which the lifetime is endogenously determined. In models like ours, the constant term b of the utility fonction plays an important and "non-standard" role because it has implications on the decisions of the agents. This point is discussed in detail by [START_REF] Rosen | The Value of Changes in Life Expectancy[END_REF] and [START_REF] Hall | The Value of Life and the Rise in Health Spending[END_REF]. In standard microeconomic theory, it is always possible to consider an increasing transformation of a utility fonction. This transformation does not impact the marginal utility ratios and thus the optimal decisions. Our model works very differently than standard ones. Indeed, in our framework, as in Rosen (1988)'s, the choice problem is a very particular one. The agent has to determine the optimal consumption and health investment plans. These decisions only depend on the marginal utility and the constant b does not impact the optimality conditions. Our model is also a life-cycle one and the agent has to determine his optimal lifetime T * . The associated optimality condition (see subsection 2.1) significantly differs from the other optimality conditions and the level of utility now matters. It is thus easy to understand that the value of b will significantly alter agent decisions (through the value of the optimal lifetime). Furthermore, the optimization problem determining the optimal lifetime has a solution if b is positive and sufficiently large.

The life-cycle utility We follow [START_REF] Rosen | The Value of Changes in Life Expectancy[END_REF] who explains how this utility function may be set and how the constant b may be understood.

The lifetime is T . The instantaneous utility of the agent (during his lifetime) is:

u(C t , H t ) = (C γ t H 1-γ t ) 1-σ 1 -σ
The agent utility over its life-cycle is given by:

∫ T 0 e -ρt u(C t , H t )dt
However, the above expression does not provide a complete evaluation of the agent utility over his life-cycle. Indeed, we suppose that the agent incurs a disutility of being death, the instantaneous disutility being -b. Evaluated at t = 0, the disutility of being death at t > T is given by β t (-b). To sum up, the life-cycle utility of the agent is written as:

∫ T 0 e -ρt u(C t , H t )dt + ∫ ∞ T e -ρt β t (-b)dt
Rearranging the above expression, one gets:

∫ T t=0 e -ρt u(C t , H t ) + ∫ ∞ T e -ρt (-b)dt = ∫ T t=0 e -ρt (u(C t , H t ) + b) dt - b ρ
The term b ρ being independent of T may be dropped. We thus obtain the intertemporal utility fonction of subsection 2.1, that is:

∫ T t=0 e -ρt (u(C t , H t ) + b) dt

Optimal longevity

The optimality condition giving the optimal lifetime T * is given in Subsection 2.1. One has:

H(T * ) = (C γ T * H 1-γ T * ) 1-σ 1 -σ + b + q T * (I T * -δ T * H T * -1 ) = 0
Evaluated at t = 0, the above condition can be rewritten as follows:

e -ρT * H(T * ) = e -ρT * [u(C T * , H T * ) + b] + e -ρT * q T * (I T * -δ T * H T * -1 ) = 0 or e -ρT * H(T * ) = e -ρT * [u(C T * , H T * ) + b] + e -ρT * q T * ḢT * = e -ρT * [u(C T * , H T * ) + b] + β T * u C (C T * , H T * ) 1 A ḢT * (A.1) = 0
It is possible to show that e -ρT * H(T * ) is the marginal net gain of being alive one more period6 . If the agent stays alive one period more, his gain in utility is e -ρT * [u(C T * , H T * ) + b], the sum of the utility derived from consumption and health capital and of the utility of being alive (b). However, in order to ensure the supplementary consumption (and maintain the health capital above or equal to the threshold H) induced by the increase of the lifespan, the agent must reduce its consumption over his life-cycle. The cost, in term of utility, is given by e -ρT * u C (C T * , H T * ) 1

A ḢT * = e -ρT * q T * ḢT * . We underline that ḢT * is necessarily negative. At the end of his life, the health capital of the agent is decreasing, so that e -ρT * q T * ḢT * < 0. At the optimum, the net gain of being alive one more period is equal to zero.

This optimality condition is written in terms of utility level, and thus, the constant term b plays a key role in the tradeoffs made by the agent. To underline the role played by b assume that b = 0. With our specification of the utility function and assuming σ > 1 (see [START_REF] Hall | The Value of Life and the Rise in Health Spending[END_REF]), the condition (A.2) becomes:

1 1 -σ + γ C T * 1 A ḢT * = 0
All the terms of the above expression are negative. It follows the optimality condition that gives the optimal lifespan works only if b > 0 is sufficiently large to ensure a strictly positive utility. In other words, the parameter b is crucial to obtain a well-posed optimization problem.

The role of parameter b Here we discuss the role of parameter b and how it determines the agent's choices. As previously mentioned, b is the utility of being alive. An increase in b will provide incentives to increase lifetime for a given amount of resources. Given the resources constraints, the agent should then reduce his consumptions. We consider a simplified version of [START_REF] Rosen | The Value of Changes in Life Expectancy[END_REF]'s model. The agent is endowed with wealth W and the interest rate is equal to the time preference rate. We thus use ρ to compute the sum of discounted consumption flows in the budget constraint. We also impose that consumption is constant over time. One has:

∫ T 0 e -ρt [ C 1-σ 1 -σ + b ] dt = 1 -e -ρT ρ [ C 1-σ 1 -σ + b ] W = ∫ T 0 e -ρt Cdt = 1 -e -ρT ρ C
Determine now the optimality conditions with respect to C and T , that is:

1 -e -ρT ρ C -σ -λ 1 -e -ρT ρ = 0 e -ρT [ C 1-σ 1 -σ + b ] -λe -ρT C = 0
λ being the Lagrange multiplier.

The optimal consumption is given by:

C = ( σ -1 σ ) 1 1-σ b 1 1-σ Note that C exists only if b > 0 and σ > 1. Let's differentiate C with respect to b: ∂C ∂b = ( σ -1 σ ) 1 1-σ 1 1 -σ b 1 1-σ -1 < 0
The optimal value of T is given by:

W = 1 -e -ρT ρ ( σ -1 σ ) 1 1-σ b 1 1-σ
It is easy to show that ∂T ∂b > 0. This example shows clearly the role of parameter b and what are the tradeoffs made by the agent. An increase in b correspond to an increase in the utility of being alive. If b increases, the agent chooses to live longer and, knowing that its wealth is given, reduces his instantaneous consumption.

To finish, we discuss what happens if the agent can invest in health capital. We use the model with a survival rate of Hall and Jones (2005), which allows to have analytical results. The agent's problem writes:

max p(H)U (C) s.t. Y -C -H ≥ 0 p(H) = H ω ,
with ω < 1 is the survival rate and Y is the agent's income.

We deduce the following optimality conditions:

p(H)U ′ (C) -λ = 0 p ′ (H)U (C) -λ = 0
Note that the level of utility must be positive, otherwise the second optimality condition is not consistent. Let's define the elasticities ξ

H = H p ′ (H) p(H) and ξ C = C U ′ (C) U (C) .
From the optimality conditions, we deduce:

C H = ξ C ξ H = ξ C ω
Assuming that ∂C ∂Y > 0, we determine the effect of an increase in Y on the ratio C H . One gets:

∂ ∂Y ( C H ) = 1 ω ∂ξ C ∂C ∂C ∂Y
The sign of the above expression depends on the sign of the derivative of the elasticity ξ C . If this elasticity is decreasing in consumption, then, the ratio C H is decreasing in Y . It follows that the share of health expenditures increases as the income Y increases.

Suppose now that U (C) = C 1-σ 1-σ + b. One has:

∂ξ C ∂C = (1 -σ)bC 1-σ ( C 1-σ 1-σ + b ) 2
Knowing that σ > 1, the above derivative is negative. It is also easy to check that if b = 0,

ξ C = 1 -σ. It follows that C H = 1-σ ω < 0 and ∂ξ C ∂C = 0.
Once again, we can underline that the importance of parameter b which must be large enough to ensure the existence of a solution to the agent's problem7 .

To conclude this paragraph about parameter b. Assuming that σ > 1, we saw, through several examples, that b must be positive and sufficiently large to ensure the consistency of a life-cycle model with endogenous life-time (or endogenous survival rate). Furthermore, b > 0 ensures that the consumption-elasticity of utility is decreasing. It follows that the share of health expenditures in income increases as income increases. This is consistent with the data providing evidences that the share of income devoted to health expenditures increases as income increases.

B The discretized system of equations

The initial health capital H 0 is supposed to be given. The system to solved is is defined by the following set of equations:

γ (C γ t H 1-γ t ) 1-σ C t -µ t = 0, t ∈ [0, T ] q t A -µ t + η t = 0, t ∈ [0, T ] -qt + (ρ + δ t )q t -(1 -γ) (C γ t H 1-γ t ) 1-σ H t -µ t θαH α-1 t -ξ t = 0, t ∈ [0, T ] η t I t = 0, t ∈ [0, T ], with η t ≥ 0 ξ t (H t -H) = 0, t ∈ [0, T [, with ξ t ≥ 0 -Ḣt + AI t -δ t H t = 0, t ∈ [0, T ] θ t i H α t i -C t i -I t i = 0, t ∈ [0, T ] q T (H T -H) = 0, with q T ≥ 0
The time interval [0, T ] is discretized in N points, that is: {t 1 , ..., t N } with t 1 = 0 and t N = T and dt = t it i-1 .

The unknown variables are: The discretized system writes as follows:

C t i , t i =
γ (C γ t i H 1-γ t i ) 1-σ C t i -µ t i = 0, t i = t 1 , . . . , t N q t i A -µ t i + η t i = 0, t i = t 1 , . . . , t N [ -(ρ + δ t i+1 )q t i+1 + (1 -γ) (C γ t i+1 H 1-γ t i+1 ) 1-σ H t i+1 + µ t i+1 θαH α-1 t i+1 + ξ t i+1
] dt +q t i+1q t = 0, t i = t 1 , . . . , t N -1 η t i I t i = 0, t i = t 1 , . . . , t N with η t i ≥ 0 ξ t i (H t i -H) = 0, t i = t 1 , . . . , t N -1 with ξ t i ≥ 0 -H t i+1 + H t i + (AI tδ t i H t i )dt = 0, t i = t 1 , . . . , t N θ t i H α t i -C t i -I t i = 0, t i = t 1 , . . . , t N q t N (H t N +1 -H) = 0, with q t N ≥ 0 There are N × 7 -1 equations and N × 7 -1 unknown variables. The system of equations is thus "well-posed" and we can undertake to solve it numerically using a Newton algorithm.

Figure

  Figure 1: Evolution of main variables

Table 1

 1

				: Benchmark calibration
	ρ	b σ	γ	α	A	δ 1	δ 2	H 0	H
	0.04 4 1.5 0.98 0.15 1 0.035 0.025 5.1558 2.7545
			Table 2: Benchmark calibration
				θ x 0	v 1	v 2
				1 0 0.0470 -0.0009

Table 3

 3

		: Elasticity σ
	σ	1.25	1.5	2
	Lifespan	89	85	81
	Health spending share 21.38% 17.17% 14.91%

Table 4

 4

		: Parameter b
	b	2	3.8	6
	Lifespan	2	85	88
	Health spending share 9.48% 17.17% 21%
	Figure

Table 5

 5

		: Parameter γ
	γ	0.9	0.98	1
	Lifespan	86	85	84
	Health spending share 21.57% 17.17% 15.46%

  t 1 , . . . , t N H t i , t i = t 1 , . . . , t N I t i , t i = t 1 , . . . , t N q t i , t i = t 1 , . . . , t N µ t i , t i = t 1 , . . . , t N η t i , t i = t 1 , . . . , t N ξ t i , t i = t 1 , . . . , t N -1

Following Halliday, He, Ning, and Zhang (2017) and[START_REF] Fonseca | Retirement and the Demand for Health[END_REF], the health index is monotonically decreasing over the life-cycle.

[START_REF] Halliday | Health Investment over the Life-Cycle[END_REF] report an hump shaped profile of consumption with an increase of

% between 25 and 50 years old and successive decreases of 30% between 50 and 60 years old and of 28% between 60 and 80 years old.

This can be shown by differentiating the agent welfare with respect to T along an optimal path.

If b = 0, the problem has a solution only if σ ∈]0, 1[ and the ratio is constant. The health spending share is then independent of income.

This research was supported by the PANORisk Regional grant (Pays de la Loire, France) 1

C Graphics