Louis Arnault
email: louis.arnault@dauphine.fr

Andreas Goltz

How would informal caregivers react to an increase in formal home-care use by their elderly dependent relatives in France? *

Keywords: JEL classification: I11, I12, J22 Long-term Care, Informal Care, Formal Care, Bivariate Tobit Model, Instrumental Variable

How would informal caregivers react to an increase in formal home-care use by their elderly dependent relatives in France?

Louis Arnault, Andreas

Introduction

In France, as in most developed countries, the population is aging and will continue to do so in the decades to come. According to the French National Institute of Statistics (INSEE), the proportion of persons aged 65 years and older reached 13.9% in 1990 and 16.6% in 2008. The preliminary estimate for January 2011 is 16.9%. In 2025, the elderly are projected to constitute 21.7% of the French population, and this figure is expected to rise to 26.2% in 2050. This aging could have many consequences for public health. Although healthy aging is now possible, population aging could increase the number of dependent elderly individuals.

An elderly dependent can be helped by his/her family or by professional caregivers, either at home or in an institution. Home health care can be formal (provided by professional caregivers) or informal (provided by one or several members of the circle of acquaintance). Most European countries encourage elderly dependent to stay home to reduce public expenditure. Given the state of the global economy, governments rely on families to care for elderly dependent. In the next few years, however, the number of informal caregivers per elderly dependent is expected to decrease (Lecroart [START_REF] Lécroart | Projections du nombre de bénéficiaires de l'APA en france à l'horizon 2040-2060[END_REF], Marbot and Roy [START_REF] Marbot | Projections du coût de l'APA et des caractéristiques de ses bénéficiaires à l'horizon 2040 à l'aide du modèle Destinie[END_REF]). In addition to demographic trends, the greater number of active seniors will make children less available to help their elderly dependent parents (Filatriau [13], Bonnet et al. [START_REF] Bonnet | La dépendance : aujourd'hui l'affaire des femmes, demain davantage celle des hommes ?[END_REF]). The smaller number of siblings and the growing physical distance between parents and children may also explain this phenomenon (Joël [22]).

In this difficult economic context, with an anticipated increase in the number of elderly dependent and a shortage of informal caregivers, a better understanding of the care arrangements of elderly dependent living at home is urgently needed to anticipate the future effects of public policies intended to reduce the cost of care and maintain efficiency. Beyond simply analyzing determinants, we aim to understand how formal and informal care interact. How would the amount of informal care vary if the amount of formal home care increased? Studying the relationship between formal informal care could help public administrators answer an important public health-policy question: is there a crowding out of familial care by public resources devoted to elderly dependent in France? Will a public subsidy helping elderly dependent pay for formal home care decrease the amount of care provided by family?

Although studies have been published on the subject for about thirty years, the question of whether informal and formal care complement or compete with each other has not been decisively answered. The most studied effect in the recent literature is that of informal care on the use of formal care. Most of the existing results indicate that informal care substitutes for formal care, after controlling for endogeneity. An older study by Greene [START_REF] Greene | Substitution between formally and informally provided care for the impaired elderly in the community[END_REF] controls for endogeneity and concludes that informal care reduces use of formal care, but the data in the study are drawn from only one American state. Lo Sasso and Johnson (2002) [START_REF] Lo Sasso | Does informal care from adult children reduce nursing home admissions for the elderly?[END_REF] and [START_REF] Charles | Can family caregiving substitute for nursing home care[END_REF] [START_REF] Charles | Can family caregiving substitute for nursing home care[END_REF] conclude that informal care reduces the risk of entering into a nursing home and thus can be seen as a substitute for institutionalization. Using the instrumentalvariable techniques, Van Houtven and Norton (2004) [START_REF] Van Houtven | Informal care and health care use of older adults[END_REF] find that informal care substitutes for formal long-term care (nursing homes and home-based care) as well as for health care (hospital and doctor visits). [START_REF] Bolin | Informal and formal care among single-living elderly in Europe[END_REF] [START_REF] Bolin | Informal and formal care among single-living elderly in Europe[END_REF] explore the same topic using European data and the strategy used by Van Houtven and Norton (2004) [START_REF] Van Houtven | Informal care and health care use of older adults[END_REF], employing child characteristics as instruments of informal care. They find that informal care acts as a substitutes for formal home care but complements doctor and hospital visits. [START_REF] Bonsang | Does informal care from children to their elderly parents substitute for formal care in Europe?[END_REF] [START_REF] Bonsang | Does informal care from children to their elderly parents substitute for formal care in Europe?[END_REF] also uses instrumental variables but distinguishes between skilled (nursing care) and unskilled (paid domestic help) formal home care. Using data from the first wave of the Survey of Health, Ageing and Retirement in Europe (SHARE) in a two-part utilization model, he finds that informal care decreases use of low-skilled home care while complementing skilled home care in Europe.

We are more concerned about the effect of formal home-care use on informal care for elderly dependents living at home in France. In 2002, the French government introduced the "Allocation Personnalisée d'Autonomie" (APA), a benefit to help elderly dependents to pay for formal home care. Three existing studies on French data examine the effect of APA on family involvement, with mixed results. First, [START_REF] Petite | Les effets de l'allocation personnalisée d'autonomie sur l'aide dispensée aux personnes âgées[END_REF] [START_REF] Petite | Les effets de l'allocation personnalisée d'autonomie sur l'aide dispensée aux personnes âgées[END_REF] compare informal care before and after the introduction of the APA in a sample of 2164 APA recipients using data collected by the French Directorate of Demographic and Social Statistics (DREES). They conclude that informal caregivers maintain their level of care; however, the elderly dependents retrospectively declared the amounts of care received. According to the authors, individuals likely underestimate the changes in care hours received. Moreover, their needs could have increased by the time they benefitted from the APA. The treatment effect of the APA is thus difficult to identify. [START_REF] Rapp | Public financial support receipt and non-medical resource utilization in alzheimers disease results from the plasa study[END_REF] [START_REF] Rapp | Public financial support receipt and non-medical resource utilization in alzheimers disease results from the plasa study[END_REF] use a cross-sectional sample of 1131 French elderly dependents affected by Alzheimer's disease and find that receiving the APA would be associated with an increase in the total number of care hours and with a lower ratio of informal to formal care, but not with a decrease in the total number of informal-care hours. Rapp's study has several limitations. First, the results apply to a specific population affected by Alzheimer's disease and using informal care. Additionally, the authors consider only the hours of informal care provided by primary informal caregivers. [START_REF] Fontaine | The effect of public subsidies for formal care on the care provision for disabled elderly people in france[END_REF] [START_REF] Fontaine | The effect of public subsidies for formal care on the care provision for disabled elderly people in france[END_REF] extends the existing French literature on this topic by using matched sampling to compare the care received by APA recipients and the care received by a control group of non-recipients. He aims to estimate both the direct effect of benefiting from the APA on formal home-care use and the indirect effect on informal-care provision and finds that publicly funded formal home care partially replaces informal care, especially for slightly disabled individuals whose informal caregivers are partners. Nevertheless, like the previous authors, he considers only APA benefits. As publicly funded formal home-care plans provided through the APA differ significantly among individuals, it is impossible to compute elasticities to quantify impacts on demand for informal care. Additionally, he finds that "the use of the APA is neither random nor exogenous", but the matching procedure that generated this result is questionable. First, it is based on a unidimensional propensity score, which is problematic because several individuals are matched despite having different observable characteristics. Second, the results that he presents do not account for the potential presence of unobserved heterogeneity. Several past studies focus on the same topic using American or European data, also with mixed results. Neither [START_REF] Christianson | The evaluation of the national long term care demonstration. 6. the effect of channeling on informal caregiving[END_REF] [START_REF] Christianson | The evaluation of the national long term care demonstration. 6. the effect of channeling on informal caregiving[END_REF], studying National long-term care in the United States in the 1980's (the Channeling), nor Motel-Klingebiel et al. (2005) [START_REF] Motel-Klingebiel | Welfare states do not crowd out the family: evidence for mixed responsibility from comparative analyses[END_REF], using data from four European countries and Israel, find any significant crowding out of informal caregivers. [START_REF] Pezzin | Does publicly provided home care substitute for family care? experimental evidence with endogenous living arrangements[END_REF] [START_REF] Pezzin | Does publicly provided home care substitute for family care? experimental evidence with endogenous living arrangements[END_REF] also use data from the Channeling experiment and find that the effect of an increase in the publicly provided home-care hours on informal-care hours is limited, after controlling for the living arrangements of the elderly dependent. By contrast, [START_REF] Viitanen | Informal and formal care in Europe[END_REF] [START_REF] Viitanen | Informal and formal care in Europe[END_REF] finds a significant crowding out of informal caregivers living outside the household after an increase in long-term care expenditure using panel data from twelve European countries, including France. Similarly, [START_REF] Stabile | Household responses to public home care programs[END_REF] [START_REF] Stabile | Household responses to public home care programs[END_REF] find that an increased availability of public home care is associated with a decline in informal care in Canada. They use instrumental variables correlated with the generosity of the public home-care program in each province. [START_REF] Golberstein | The effect of medicare home health care payment on informal care. Inquiry: a journal of medical care organization[END_REF] [START_REF] Golberstein | The effect of medicare home health care payment on informal care. Inquiry: a journal of medical care organization[END_REF] note that the data used by [START_REF] Stabile | Household responses to public home care programs[END_REF] [START_REF] Stabile | Household responses to public home care programs[END_REF] are limited in only having information on whether informal care is delivered. Moreover, the true exogeneity of their instrument for formal care, i.e., the generosity of the public home-care program per province, is questionable. Using longitudinal data from AHEAD and HRS surveys, [START_REF] Golberstein | The effect of medicare home health care payment on informal care. Inquiry: a journal of medical care organization[END_REF] [START_REF] Golberstein | The effect of medicare home health care payment on informal care. Inquiry: a journal of medical care organization[END_REF] find that informal care increases for individuals exposed to more restrictive payment caps on Medicare home health care.

Although it could have major implications for future public policy, the effect of formal home care on informal care for elderly dependents living at home has not been studied as often as the reverse. To the best of our knowledge, [START_REF] Stabile | Household responses to public home care programs[END_REF] [START_REF] Stabile | Household responses to public home care programs[END_REF], using Canadian data, wrote the only study suggesting an instrumental variable model. Instruments for measuring formal home care, however, are difficult to find and subject to controversy. In this study, we extend previous findings in several different directions: first, we study the effect of formal home care on informal care using French 2008 household Disability -Healthcare data (Handicap Santé Ménages -HSM 2008). Second, in contrast with [START_REF] Stabile | Household responses to public home care programs[END_REF] [START_REF] Stabile | Household responses to public home care programs[END_REF], we use hours of informal care received by each elderly dependent as a quantitative variable. Third, we extend the classical two-part model in a bivariate Tobit model to account for the censor of our formal home-care variable. Fourth, we use a new instrument (or exclusion variable) adapted to France to account for endogeneity of formal home care.

2 Background and conceptual framework

Relationship between formal home care and informal care

Theoretical models related to the utilization of formal and informal care among the elderly are mainly based on family decision-making and health production, where formal home care and informal care are regarded as two factors of production. The model described by Van Houtven and Norton (2004) [START_REF] Van Houtven | Informal care and health care use of older adults[END_REF] is an extension of the classic [START_REF] Grossman | On the concept of health capital and the demand for health[END_REF] [START_REF] Grossman | On the concept of health capital and the demand for health[END_REF] model of health demand, altered to include formal and informal care. The relationship between informal and formal care depends on the sign of the derivative of the marginal product of formal care (in the production of health) with respect to informal care. According to [START_REF] Bolin | Informal and formal care among single-living elderly in Europe[END_REF] [START_REF] Bolin | Informal and formal care among single-living elderly in Europe[END_REF] and [START_REF] Bonsang | Does informal care from children to their elderly parents substitute for formal care in Europe?[END_REF] [START_REF] Bonsang | Does informal care from children to their elderly parents substitute for formal care in Europe?[END_REF], complementarity or substitution between formal home care and informal care is an empirical issue. The decision to provide informal care to an elderly dependent parent and the dependent's decision to ask for formal care are simultaneously determined. In this article, we focus only on the effect of formal home care on informal care, which has been little studied before now and is still subject to mixed results.

Two main hypotheses are tested in this article (cf. Van Houtven and Norton (2004) [START_REF] Van Houtven | Informal care and health care use of older adults[END_REF]). First, we examine the nature of the empirical relationship between formal and informal care. This relation is not straightforward. Contrary to expectations, formal home care and informal care are not necessarily mutually exclusive. They may be complementary if, for example, formal home care consists more in personal care (like bathing) and informal care more in domestic tasks. Several normative or emotional considerations can also have an impact on the degree of participation by family members. An informal caregiver may say, for example: "I help 1 hour a day, whatever the other quantities of care my parent receives". Such an arrangement would imply an absence of substitution between formal and informal care. We then test whether the effect of formal home care on informal care is likely to differ according to the type of formal home care used. In the manner of [START_REF] Bonsang | Does informal care from children to their elderly parents substitute for formal care in Europe?[END_REF] [START_REF] Bonsang | Does informal care from children to their elderly parents substitute for formal care in Europe?[END_REF], we isolate personal formal home care (like bathing) and expect the substitution effect between formal home care and informal care to be lower than for other kinds of formal home care, such as paid domestic help.

Specificities of the French case

In France, formal home care can be provided to elderly dependents in several ways. Service providers of formal home care hire and pay employees to care for elderly dependents. As service providers engage in a regulated activity, they are subject to agreements provided by French public agencies. We can distinguish service providers agreed to by French District Councils from those agreed to by French Regional Offices for Labor. The former cannot choose their prices, while the latter have more flexibility as long as their prices do not vary dramatically from one year to another. An elderly dependent can also pay the care provider directly by recruiting over the counter. The spectrum of formal home care providers is thus very different from one French district to another. Moreover, visiting nurses and housekeepers can practice anywhere in France. As a result, there is a significant imbalance between districts in terms of supply. In addition, in France as in many European countries, the out-of-pocket expenses for an elderly dependent receiving formal home care are reduced thanks to the "Allocation Personnalisée d'Autonomie" (APA). Since 2002, this benefit has been allotted by the French District Councils. To benefit from the APA, people fill out an application. Each district has its own application, with varying levels of complexity and numbers of supporting documents. Out-of-pocket expenses for formal home care can vary widely from a district to another.

To sum up, there is substantial variability among French districts in terms of access to formal home care and formal home-care providers [START_REF] Gramain | Réagencements territoriaux et conduites des politiques sociales à l'échelle locale[END_REF] [START_REF] Gramain | Réagencements territoriaux et conduites des politiques sociales à l'échelle locale[END_REF]). These differences should have an impact on individual demands for formal home care. We thus must take them into account to build a strong instrumental variable for formal home care.

Estimation strategy: A bivariate Tobit model

The aim of this work is to analyze the causal effect of formal home care provided to elderly dependents on the quantity of informal care received. To test the two hypotheses mentioned in the Subsection 2.1, we estimate a bivariate Tobit model, which resembles the well-known IV Tobit model, except that our formal-care variable in the structural equation is censored. The first (structural) equation has the form of a Tobit model that predicts the (logged) hours of informal care received (y), which is censored at 0. The second (instrumental) equation takes the form of a Tobit model explaining the (logged) hours of formal home care received (f c, censored at 0) by exogenous variables and an exclusion variable (Z, also named instrument). We use the density of self-employed midwives in each district as an exclusion variable. The two error terms follow a bivariate normal distribution. The utilization of informal care (y) is a function of formal home care (f c) and of a vector of exogenous characteristics of the individual (X). The subscript i represents the individual. Our bivariate Tobit model can be written as follows:

y * i = γ 0 + γ f c ln(1 + f c i ) + γ X X i + s f c * i = δ 0 + δ Z Z i + δ X X i + f with: ln(1 + y i ) = y * i , if y * i > 0 0, else. and: ln(1 + f c i ) = f c * i , if f c * i > 0 0, else.
where y * and f c * are latent variables (only observed when they take positive values) related to informal care and formal home care, respectively, and γ 0 , γ f c , γ X , δ 0 , δ X are the parameters (or vectors of parameters) to be estimated. Because the formal home-care and informal-care variables (f c and ic) are skewed, we take the log in our model. This bivariate Tobit model is simultaneously estimated with Stata 11 using maximumlikelihood techniques. The log-likelihood is easily calculated following Amemiya (1974) [START_REF] Amemiya | Multivariate regression and simultaneous equation models when the dependent variables are truncated normal[END_REF], [START_REF] Lollivier | Econométrie avancée des variables qualitatives[END_REF] [START_REF] Lollivier | Econométrie avancée des variables qualitatives[END_REF] and [START_REF] Fontaine | The trade-off between informal care and work in Europe. Le soutien familial aux personnes âgées dépendantes[END_REF] [START_REF] Fontaine | The trade-off between informal care and work in Europe. Le soutien familial aux personnes âgées dépendantes[END_REF]. The explicit version of the log-likelihood is available upon request.

A two-part IV model could also be well fitted to answer the question (cf. Van Houtven and Norton (2004) [START_REF] Van Houtven | Informal care and health care use of older adults[END_REF] and [START_REF] Bonsang | Does informal care from children to their elderly parents substitute for formal care in Europe?[END_REF] [START_REF] Bonsang | Does informal care from children to their elderly parents substitute for formal care in Europe?[END_REF]). In our bivariate Tobit model, however, the endogenous formal homecare variable (f c i ) is censored (and treated as well), as there is a large number of individuals (more than 31 % of our population of interest) for whom this variable equals 0. This variable has a large mass point in 0, and the two-part IV model does not take it into account. In contrast to Heckman's selection model, the probability of receiving informal care and the amount of informal care received are independent; this very strong assumption should be relaxed. These problems shift our focus to the bivariate Tobit model. Nevertheless, we also want to model the decision to receive informal care separately from the quantity received. Consequently, a two-part IV model is presented in Appendix A.1 and estimated in Appendix A.2.

Data

We use the 2008 household Disability -Healthcare data (Handicap Santé Ménages -HSM 2008) of the French National Institute for Statistics and National Studies (INSEE). The aim of this survey is to acquire as much information as possible about care-dependent people in France; it thus includes both information about health status, socio-economic status, living situation and care received. In total, 29,931 individuals answered the questionnaire.

Sample selection criteria

The aim of our work is to describe how elderly people in France are cared for. We define elderly people as being at least 60 of age and exclude younger care-dependent people.

We also exclude all completely autonomous people. We use a very broad definition of dependence to avoid excluding individuals, who are only slightly care-dependent. We base our definition of dependence on the number of difficulties in activities of daily living (ADL) or instrumental activities of daily living (IADL). The ADL include fundamental tasks necessary for an individual to live and survive alone. IADL are not necessary to survive but enable the person to live alone. Seven ADL are taken into consideration in our work: bathing, dressing and undressing, cutting food, eating and drinking, using the toilet, lying down and getting up from bed, and sitting down in and getting up from a chair. Eleven IADL are considered: shopping, preparing meals, doing common household chores, doing less common chores, doing common administrative processes, taking medication, moving around, leaving home, using transportation, finding the way, and using a telephone. For each ADL and IADL, the individuals are asked the following question: "How much difficulty do you have...". People are considered dependent if they report having at least "some difficulty" in at least one ADL or IADL. They are also considered care-dependent if they suffer from Alzheimer's disease.

Moreover, informal caregivers living with the care recipient often have difficulty declaring the exact amount of time they spend caring for their cohabitant. There is thus a high proportion of missing values for this particular group of caregivers. The amounts of care declared are also not as robust as the numbers given by informal caregivers living somewhere else. The distinction between care and regular household duties is sometimes very difficult to make. Many spouses seem to consider any caring tasks as marital duties and consequently report a very small number of care hours, even if they actually help much more. A similar idea is developed by [START_REF] Paraponaris | L'utilisation de l'évaluation contingente pour valoriser l'aide informelle apportée aux personnes souffrant de handicap ou en perte d'autonomie : quelle intelligibilté dans le[END_REF] [START_REF] Paraponaris | L'utilisation de l'évaluation contingente pour valoriser l'aide informelle apportée aux personnes souffrant de handicap ou en perte d'autonomie : quelle intelligibilté dans le[END_REF], who posit that caregivers have less chance to correctly evaluate the cost of an hour of care when they provide it to a close relative than to a more distant one. Information on the health status of the spouse is also not available in our data. Assuming that a significant proportion of spouses are themselves in need of care, we cannot distinguish them from those able to help their partner. We have thus decided to include only care-dependent people living alone.

We also discard observations with missing or unreliable values for the variables of interest and the other explanatory variables; ultimately, 1526 individuals meet all the criteria.

Formal home-care and informal-care variables

Our variable of interest is the weekly hours of informal care received by the respondent. This amount is aggregated and is defined as the sum of care hours provided by all informal caregivers per week. For each informal caregiver, the respondents have three possibilities: they can answer in terms of hours per day, per week or per month. We transform each answer into hours per week. We focus on "physical care", which means that financial help and moral support are not taken into account. The formal home-care variable consists in the weekly hours of formal home care received by the respondent. Like informal care, this amount of formal home care received is aggregated and is defined as the amounts of care provided by each formal caregivers, transformed into hours per week.

Later in our article, we isolate personal formal home care from the total amount of formal home care. For a given elderly dependent, we possess only information about the amount of formal home care provided by each formal caregiver and why he or she receives help from this particular caregiver. A given formal caregiver can help an individual perform several very different tasks (personal care and domestic chores for example). We cannot know how much time he or she spends performing each of these tasks. Thus, the distinction between personal care and other kinds of formal home care is made according to the profession of each formal caregiver. Formal home care provided by a nurse, a nurse's aide, another paramedical professional or a psychologist/psychomotor therapist is assumed to be personal care. It is harder to consider the remaining modalities as belonging to another identified formal home-care group for two reasons. First, several personal formal home caregivers (such as personal-care assistants) cannot be isolated and belong to this group. Second, this group lumps social caregivers with housekeepers providing paid domestic help. Third, we do not really know what the modality "other professionals" includes.

The exclusion variable

Thanks to the censor of our care variables, the parameters of the bivariate Tobit are theoretically identifiable, even without an exclusion variable Z in the formal home-care equation. Empirically, it is better if identifiability does not rest only on the censor of both care variables [START_REF] Lollivier | Econométrie avancée des variables qualitatives[END_REF] [START_REF] Lollivier | Econométrie avancée des variables qualitatives[END_REF]). The success of our estimation hinges on finding a good exclusion variable (or instrument) for formal home care, which means a variable highly correlated with formal home care but not correlated with the error term in the equation of informal care. It is difficult to find such a variable, as the effect of formal home care use on informal care provision is not studied as often as the reverse. To our knowledge, only [START_REF] Stabile | Household responses to public home care programs[END_REF], [START_REF] Stabile | Household responses to public home care programs[END_REF], using Canadian data, have suggested an instrumental variable model. Their instruments for formal home care are correlated with the generosity of the public home-care program: the proportion of the population aged 65 and older in each province, the level of provincial spending on education in each province and the provincial tax rate as a share of federal taxes in each province. As discussed in Section 1, their exogeneity is subject to controversy (cf. [START_REF] Golberstein | The effect of medicare home health care payment on informal care. Inquiry: a journal of medical care organization[END_REF] [START_REF] Golberstein | The effect of medicare home health care payment on informal care. Inquiry: a journal of medical care organization[END_REF]).

We use a new instrument, which is assumed to be more adapted to the French situation, aiming to catch differences between French Council Districts, which can explain the level of formal home care received. The exclusion variable must catch the attractiveness of the territory for professional caregivers. Because visiting nurses or housekeepers can practice their job anywhere, in France, as in many European countries, there is a geographical imbalance among districts in terms of formal home-care supply. [START_REF] Genet | Home care across europe[END_REF] [START_REF] Genet | Home care across europe[END_REF] explain that differences in the density of the home-care network across Europe and within European countries influence access to care. In attractive areas, the greater number of formal home caregivers may give better access to formal home care for elderly dependents and increase demand.

Moreover, the risk of induced demand effects is greater in attractive areas. [START_REF] Delattre | Fixed fees and physician-induced demand: A panel data study on french physicians[END_REF] [10] study induced demand as applied to French physicians. A high density of physician increases competition and thus favor-induced demand. To maintain or increase their income despite competition, physicians advise patients to consume large volumes of health care. The same argument can be advanced for formal home care suppliers. If the formal home care supply is greater in attractive territories, then formal home caregivers have to find strategies to attract loyal "customers" in order to remain competitive. It also tends to increase the demand for formal home care through induced demand.

Although the French benefit for autonomy (APA) is well-known by a large part of elderly dependents, there are still people who do not know of its existence, especially in low-income groups (GRATH survey report (2009) [START_REF]Groupe de Réflexion et Réseau pour l'Accueil Temporaire[END_REF]). In attractive areas, the global demand for formal home care should be greater, partly because of the induced demand previously mentioned. Elderly dependents living in such areas have more chances to interact with others who use formal home-care services. It is thus easier for an elderly dependent living in an attractive area to access information about formal home-care supply and/or benefits. Better access to information should increase in return the use of formal home care in attractive areas.

It would have been interesting to introduce the density of visiting nurses or of housekeepers in each district as a potential instrument for formal home care. This variable would be correlated with the attractiveness of each district. Nevertheless, endogeneity might have been a problem. Indeed, high individual demand for formal home care can have a positive effect on the number of visiting nurses in the district. We introduce the number of self-employed midwives per woman aged between 15 and 50 in each district, which is also correlated with the attractiveness of the area and is exogenous in our context. Empirically, there is no effect of our instrument on either the probability of receiving informal care or on the amount of informal care received when informal care is given. The density of self-employed midwives by French district in 2007 is obtained thanks to data from the French Directorate of Demographic and Social Statistics (DREES). This variable is district-level, which means that the same value is attributed to every individuals living in the district. We take this point into account in clustering the error term of each model by district.

Later in the paper, we check the robustness of our estimates in testing another exclusion variable (instrument) in our bivariate Tobit model. A variable related to weather patterns in each district would be fully exogenous in context and highly correlated with the attractiveness of each geographical area. For example, formal home-care professionals prefer living and practicing in sunny districts than in rainy ones. [START_REF] Delattre | Stratégies de localisation des médecins généralistes français: mécanismes économiques ou hédonistes[END_REF] [START_REF] Delattre | Stratégies de localisation des médecins généralistes français: mécanismes économiques ou hédonistes[END_REF], studying the factors determining the settlement of general practitioners in France, use the average annual hours of sunshine per district as an instrument for attractiveness of the area; we extend it to explain the individual demand for formal home care. The variable is constructed using data from Météo France between 1991 and 2011.

Explanatory variables

We group the explanatory variables into several classes. The group of variables related to the elderly person's health and dependence includes Alzheimer's disease status and a dependence score. To build our score, we selected each of the seven ADL but only six IADL, excluding those that are highly correlated with the Alzheimer's variable. For each ADL or IADL, the score is increased by one if the individual reports having " some difficulty " in executing it alone, by two if he or she reports having "great difficulty" in executing it alone and by three if he or she cannot execute it alone. Each individual's dependence score is the sum of his or her values for each ADL-IADL. The group of variables related to socio-economic characteristics include having a diploma and income group (a categorical variable in five modalities). A third group of variables concerns children as potential informal caregivers. The numbers of daughters and sons are introduced as continuous variables. We could focus only on children living close to their dependent parents, as the probability of their caring for the parent is greater; however, the variable "geographical proximity" is potentially endogenous (cf. [START_REF] Charles | Can family caregiving substitute for nursing home care[END_REF] [START_REF] Charles | Can family caregiving substitute for nursing home care[END_REF], [START_REF] Bonsang | Does informal care from children to their elderly parents substitute for formal care in Europe?[END_REF] [START_REF] Bonsang | Does informal care from children to their elderly parents substitute for formal care in Europe?[END_REF]): the child may move closer to his or her dependent parent if the number of tasks that he or she has to execute for his or her parent increases. To avoid endogeneity bias, we exclude this variable. Control variables include the age and gender of the elderly dependent.

Table 1 gives summary statistics (means) for the entire sample (second column) and for four subsamples built according to the kind of care used by elderly dependents (last four columns). The third column gives summary statistics for our sample when all observations are weighted to reflect the general population. These estimates are very close to those obtained without weighting data, indicating that the sample is representative of the population of elderly dependents aged 60 and over living alone at home in France. Henceforth, we focus exclusively on non-weighted estimates. The average age of all of the individuals in our sample is 79. While the average age of non-users is only 74, older people are overrepresented in the sample of users of both formal and informal care. At first sight, it is surprising that 82% of the individuals in the data sample are women, but we are only looking at elderly individuals who do not live with a spouse; women have a longer life expectancy and tend to be younger than their partners, increasing the proportion of women in our sample. The individuals have a mean of slightly more than two children, with marginally more sons than daughters. Individuals with children are overrepresented in subsamples of people receiving informal care and underrepresented in subsamples of non-users or of people receiving formal care only. This pattern is stronger for individuals with a daughter than for those with a son and highlights the role of children, especially daughters, in providing informal care to their elderly dependent parent living at home without spouse. On average, the dependence score is 8.743. On a scale of 39, this value is low. A substantial number of elderly dependents living alone at home are not highly care-dependent. As expected, the average dependence score for non-users is very low (2.304) but reaches 13.893 for individuals receiving both kinds of care. The level of dependence is positively correlated with the use of care. A total of 5.7% of our sample suffers from Alzheimer's disease, the main cause of mental dependence. These individuals are strongly underrepresented in subsamples of non-users and people receiving only formal home care. People suffering from Alzheimer's disease prefer informal to formal care but are overrepresented in the subsample of people receiving both kinds of care, underlining the major role of both family and professionals in caring for people with mental dependencemental dependence . The overall proportion of individuals receiving formal home care is greater than that of those receiving informal care (approximately 69% against 42 %). Again, we focus on individuals living without their spouses, who are often the main providers of informal care.

Table 2 provides the average amounts of care received by all care users (second column) and for three subsamples according to the kind of care used (last three columns). Among the care users, individuals receive on average more than 8 hours of formal home care per week and more than 7 hours of informal care. Individuals using both types of care receive on average more formal home care and more informal care than those using one kind of care exclusively. These findings are consistent with previous results Table 1: highly dependent persons need more of both formal and informal care. Figure 1 provides the Kernel density estimation of formal and informal care variables (for care users only), using the Epanechnikov Kernel. The estimated formal home-care density function shows that among formal home-care users, a large proportion of individuals receive less than 20 hours a week of care. The density function does not exhibit an upper tail. For informal-care users, the number of care hours differs widely among individuals and can be very large. Consequently, the informal-care density function exhibits a large upper tail. If both estimated density functions are far from being normally distributed, the density function of their logs approaches that of the Gaussian distribution (see Figure 2). We thus prefer working with the logs of both care variables in our models. The association between both care variables gives an initial sense of the nature of the relationship between formal and informal care in our sample. By far the most familiar measure of association (dependence) is the Bravais-Pearson's linear correlation between two variables Y 1 and Y 2 . This indicator is only powerful when the link between Y 1 and Y 2 is linear. An alternative measure of non-linear association is the rank correlation, one main indicator of which is Spearman's correlation coefficient (ρ), defined as the Pearson's correlation coefficient of the ranked Y 1 and the ranked Y 2 . Like the linear correlation coefficient, its values vary between -1 and 1, with 1 indicating perfect concordance. The Spearman's rank correlation coefficient is less sensitive than the Pearson correlation to strong outliers. Such a coefficient is calculated for both care variable in Table 3. In the global sample, the Spearman's correlation coefficient ρ is significantly negative at a 5% level but close to 0 (-0.05). Without controlling for the level of dependence, the highly dependent individuals are most likely to receive large quantities of formal home care and are highly likely to receive substantial informal care. Thus, Spearman's correlation coefficient tends to be positive. As expected, after controlling for the level of dependence1 Spearman's correlation coefficient is strongly negative and significant (at a 1% level). Formal home care and informal care are thus negatively associated, after controlling for the level of dependence.

Results and Interpretation

In this section, we present the results of the bivariate Tobit model and estimate it, isolating personal formal home care. We then check the robustness of our estimates, modifying first the exclusion variable and then the dependence score.

The bivariate Tobit model

The estimates of the bivariate Tobit model can be found in Table 4. Our instrumental variable, the density of self-employed midwives by district, is positively correlated with the hours of formal home care received and is significant at a 1% level. The more attractive the area, the more formal home care the elderly dependent uses. We try to include our instrument in the informal-care equation, even if the model is then identified only by the censor of both dependent variables. As expected, the coefficient is not significant at a 10% level, indicating that our instrument is not correlated with informal care. Formal and informal care are both positively influenced by the level of dependence. Parameters associated with the dependence score, Alzheimer's disease status and the age group are significant at the 5% level, even if Alzheimer's disease has a significant impact only on informal care. People suffering from early-stage Alzheimer's disease must be over-represented in our sample of Alzheimer's patients living alone at home. These people may not need help beyond family support, or their family may believe that to be so. People suffering from Alzheimer's disease find it difficult to collect the necessary paperwork to benefit from formal home care. The fact of having a diploma has a positive impact on formal home-care use: educated individuals may find information about the formal home-care market more easily and have easier access to it. Moreover, the fact of having a diploma can be considered a proxy variable for social class. Individual belonging to high social classes have more chance to hire an housekeeper to do domestic tasks. This effect is consistent with estimates from several studies, such as that by Van Houtven and Norton (2004) [START_REF] Van Houtven | Informal care and health care use of older adults[END_REF]. Use of informal care increases with the number of children, consistent with intuition and the literature on the subject [START_REF] Bolin | Informal and formal care among single-living elderly in Europe[END_REF] [START_REF] Bolin | Informal and formal care among single-living elderly in Europe[END_REF], Charles and Sevak (2005) [START_REF] Charles | Can family caregiving substitute for nursing home care[END_REF]) because children are the main informal-care providers for elderly dependents living without a partner. The formal home-care variable has a significant negative effect on the hours of informal care received in the bivariate Tobit model. Receiving more formal home care reduces the probability of receiving informal care and/or the quantity of informal care received when informal care is given. The model predicts a crowding out of informal caregivers when hours of formal home care increase. This model confirms the hypothesis of a substitution effect between formal and informal care, as informal caregivers are crowded out when the quantity of formal home care received increases. A two-part IV model is estimated in Appendix A.2, yielding the same conclusion: informal caregivers are crowded out when the quantity of formal home care received increases. Moreover, our explanatory variables have similar effects on the probability of receiving informal care and on the quantity of informal care received when informal care is given. This finding thus validates our choice to highlight the results of the bivariate Tobit model, which does not use this division.

Elasticities

In the previous subsection we show that the bivariate Tobit model confirms the existence of crowding out of informal caregivers, when formal home care increases. To quantify this effect, we compute average elasticities of informal care with respect to formal home care. The way we proceed to compute these average elasticities is described in Appendix B. Following DiCiccio and Efron (1996) [START_REF] Diciccio | Bootstrap confidence intervals[END_REF] we estimate bias-corrected bootstrapped confidence intervals, performing 500 bootstrap replications. The elasticity for the bivariate Tobit model is presented in Table 5. The elasticity reaches -0.425 and is significant at a 5% level. A 10% increase in formal home-care hours would lead to a 4.25 % decrease in the quantity of informal care received. Elasticities are calculated for the two-part IV model in Appendix A.3, Table 13 and are also negative and significant at a 5% level. There is a non-negligible crowding out of informal caregivers when formal home care increases. The point now is to try to understand how this effect changes if we only consider personal formal home care. For * : significant at the 10% level, * * : significant at the 5% level. We report the bias-corrected bootstrapped confidence intervals.

a given elderly dependent, we possess only information about the amount of formal home care provided by each formal caregiver and the reasons why the dependent receives help from this particular caregiver.

A given formal caregiver can help an individual perform several very different tasks (personal care and domestic chores for example). We cannot know how much time the dependent spends performing each of these tasks, so the distinction between personal care and other kinds of formal home care is made according to the profession of each formal caregiver. Formal home care provided by a nurse, a nurse's aide, another paramedical professional or a psychologist/psychomotor therapist is assumed to be personal care. It is harder to consider the remaining modalities as belonging to another identified formal home-care group for two reasons. First, several personal formal home caregivers (such as personal-care assistants) cannot be isolated and belong to this group. Second, this group lumps social caregivers with housekeepers providing paid domestic help. Third, we do not really know what the modality "other professionals" includes; thus, we only re-estimate the bivariate Tobit model with hours of personal formal home care replacing total hours of formal home care. The estimates of the bivariate Tobit model can be found in Table 6 (the re-estimated two-part IV model is presented in Appendix A.3, Table 12). The coefficient of our exclusion variable, the density of self-employed midwives, remains positive and significant at a 1% level. It is difficult to compare the coefficients of the new model with those of the previous bivariate Tobit model (with total hours of formal home care), but the main explanatory variables remain significant at a 5% level. Dependence-associated variables (dependence score, age, Alzheimer's disease status) significantly explain both care variables. Having children significantly increases the quantity of informal care received. These results are consistent with those obtained by [START_REF] Bonsang | Does informal care from children to their elderly parents substitute for formal care in Europe?[END_REF] [START_REF] Bonsang | Does informal care from children to their elderly parents substitute for formal care in Europe?[END_REF]. Education level no longer explains the quantity of personal formal home care received, and as in [START_REF] Bonsang | Does informal care from children to their elderly parents substitute for formal care in Europe?[END_REF] [START_REF] Bonsang | Does informal care from children to their elderly parents substitute for formal care in Europe?[END_REF], using formal home care for domestic tasks is more frequent among but not exclusive to highly educated individuals, as most informal caregivers are reluctant to perform such tasks. The elasticity of this new model with personal formal home care alone is computed and presented in Table 7. The elasticity is negative and significant at a 5% level, which suggests that there is crowding out of informal caregivers when personal formal home care increases, but the computed elasticities of informal care with regard to personal formal home care are much lower than elasticities of informal care with regard to total formal home care (see Table 5). Thus, according to the bivariate Tobit model estimates, a 10% increase in hours * : significant at the 10% level, * * : significant at the 5% level. We report the bias-corrected bootstrapped confidence intervals.

of personal formal home care leads only to a 1.28% reduction in hours of informal care received, while a 10% increase in total hours of formal home care leads to a 4.25% reduction in hours of informal care received. The same trend is seen when we compute the new elasticities associated with the IV two-part model (see Appendix A.3, Table 14). As expected, the crowding out of informal caregivers is lower when only personal formal home care increases. [START_REF] Bonsang | Does informal care from children to their elderly parents substitute for formal care in Europe?[END_REF] [4] obtains a similar result studying the inverse causality of informal care in the equation for formal home care. The effect of informal care on formal home-care use is greater for domestic formal home care than for personal/nursing formal home care.

Sensitivity analysis

To check the robustness of our model, we estimate our bivariate Tobit model using another exclusion variable (instrument) in the formal home-care equation. Following [START_REF] Delattre | Stratégies de localisation des médecins généralistes français: mécanismes économiques ou hédonistes[END_REF] [START_REF] Delattre | Stratégies de localisation des médecins généralistes français: mécanismes économiques ou hédonistes[END_REF], we use the average annual hours of sunshine per district. This variable seems to be fully exogenous and highly correlated with the attractiveness of each geographical area and thus with the individual demand for formal home care. The estimates of the bivariate Tobit model with formal home care using such an exclusion variable appear in Appendix A, Table 15. Our sunshine instrument is a strong predictor of formal home care; its coefficient is positively significant at a 5% level, but it does not impact the utilization of informal care at a 10% level. The results of the same bivariate Tobit model with personal formal home care only are available upon request. Elasticities can be calculated for each of these two models. Their values can be found in Table 8. * : significant at the 10% level, * * : significant at the 5% level. We report the bias-corrected bootstrapped confidence intervals.

Both elasticities are negative and significant at a 5% level, with values very close to those obtained with our first instrumental variable (-0.399 against -0.425 and -0.130 against -0.128). This result confirms the crowding out of informal caregivers and highlights that this effect is much weaker when only personal formal home care increases. We then want to know if our estimates are sensitive to the use of another variable of dependence. We estimate our bivariate Tobit model using two variables measuring dependence in both equations instead of our dependence score. We use the well-known Katz index of dependence, which is often used to evaluate the level of dependence of elderly dependents in the United States. The original index has 8 modalities, but we group them into 4 , with each catching enough individuals for statistical significance: "being weakly dependent", "being moderately dependent", "being highly dependent", and "being very highly dependent". This index is based exclusively on limitations in activities of daily living (ADL). As limitations in instrumental activities of daily living must be taken into account to correctly measure the dependence of an elderly individual, we couple the Katz index with a score of IADL limitations. The results of the new bivariate Tobit model are presented in Appendix A, Table 16. The Katz index of dependence is only significant in the formal home-care equation, whereas the IADL dependence score is significant in both. The use of informal care would be greater for individuals with limitations in instrumental activities of daily living, such as grocery shopping, than for those with limitations in "fundamental" activities of daily living, such as bathing, consistent with the intuitive outcome. The results of the same bivariate Tobit model with personal formal home care only are available upon request. Elasticities can be calculated for each of these two models. Their values can be found in Table 9. * : significant at the 10% level, * * : significant at the 5% level. We report the bias-corrected bootstrapped confidence intervals.

Both elasticities are negative and significant at a 5% level. If their values are a little bit lower than those obtained with the other dependence variable (-0.270 against -0.425 and -0.074 against -0.128), the estimates still confirm the crowding out of informal caregivers and highlight that the crowding out is much weaker when only personal formal home care increases.

Conclusion

A simple empirical model is used to better understand how French informal caregivers would react if the quantity of formal home care received by their elderly dependent relative increased. Three main results can be highlighted in this study.

First, variables related to the attractiveness of each French district are strong predictors of individual's demand for formal home care. Several factors could explain this result. The greater number of formal home caregivers in attractive areas may give better access to formal home care for elderly dependents and increase demand. Competition could drive formal home-care professional practicing in such attractive areas to find strategies to attract new "customers", increasing demand. Additionally, elderly dependents living in such areas would be better informed, as they would have more opportunities to interact with professionals or other formal home-care users.

Second, there is a crowding out of informal caregivers when formal home care increases. Third, the crowding out of informal caregivers is much lower when formal home care consists in personal care than when it consists in domestic tasks or social care. Informal caregivers are not suited for personal care or are less ready to take on some of these tasks, such as bathing their parent. In contrast, the substitution effect of informal caregivers is high for other kinds of formal home care, which mainly consist in paid home help from housekeepers or social care from social workers. This high substitution effect raises the societal and philosophical question of the role of the French public administration in long-term care. Should a benefit for formal home care from the French public administration be offered only to elderly dependents who cannot receive informal care from their family? In other terms, should the French public administration play a role subsidiary to the family structure, or does the French public administration have to help every elderly dependent without restriction? On the one hand, the subsidiary position is controversial: it would break the natural link between a family caregiver and his or her elderly dependent relative. If individuals benefiting from a familial support are not helped publicly, then their family caregivers are forced to care for them; it changes from a deliberate choice to a moral duty. On the other hand, the universal benefit is also controversial; it would encourage family caregivers to give up their role, taking advantage of the benefit granted to their elderly relative.

This societal question must be answered a priori, apart from any economic consideration. Nevertheless, one could try to understand which answer is most acceptable, economically speaking. In cost-effectiveness and cost-utility analyses, we could extend our study to better understand what this crowding out of informal caregivers really means. The French benefit for autonomy (APA) has been set up with two main goals. First, it should give individuals without informal caregivers access to care. Thus, such crowding out of informal caregivers could be interpreted as a disengagement, even more if they do not return to the labor market. Such crowding out of informal caregivers should be fought. The most socially and economically efficient option for the French public administration could be to play only a subsidiary role to the family structure. Another idea behind the APA is risk mutualization. Informal caregivers do not have to bear the cost of care alone, as this cost can be too high for them. The French benefit for autonomy also helps them to compensate for the financial loss of caring for their elderly dependent parent. Thus, if the crowding out allows overburdened informal caregivers to feel better and to return to work (cf. [START_REF] Fontaine | The trade-off between informal care and work in Europe. Le soutien familial aux personnes âgées dépendantes[END_REF] [START_REF] Fontaine | The trade-off between informal care and work in Europe. Le soutien familial aux personnes âgées dépendantes[END_REF]), then there is a social value to sharing risks. The crowding out of informal caregivers could be advantageous to individuals and society, in which case a universal formal home-care benefit could be economically and socially efficient.

We can now focus on the structural equations predicting informal care (IV Probit) and the quantity of informal care received, when informal care is given (IV Regress). The probability of receiving informal care and the quantity of informal care received are both significantly positively influenced by the level of dependence. Parameters associated with the dependence score, Alzheimer's disease status and age group are significant at the 5% level. Number of daughters is also a strong predictor of receiving informal care and of the quantity received, as daughters help their parents more than sons. The hours of formal home care have a significant negative impact on informal care in both equations. More formal home care reduces both the probability of receiving informal care and the quantity of informal care received when informal care is given. The model predicts crowding out of informal caregivers when hours of formal home care increase.

Moreover, the explanatory variables in our model seem to have the same effect on the probability of receiving informal care as on the quantity of informal care received when informal care is given. Thus, a two-part model does not provide additional information. The Wald and Wu-Hausman tests nearly reject exogeneity of formal home care in each part of the model. Thus, we compute the same two-part model without assuming that formal home care is endogenous to informal care. This model can be found in Table 11. The coefficients related to hours of formal home care are also both negative and significant at a 5% level. It is still reasonable to consider the IV two-part model as the model of reference for two reasons. First, there is a growing literature showing the effect of informal care on formal home care. Second, it is highly plausible that several unobserved characteristics of the individual, such as the geographical proximity of informal caregivers or wealth of the elderly dependent and his/her family, influence the use of both formal and informal care.

A.3 Elasticities associated with the IV two-part model

In Table 13, we calculate elasticities for both parts of our two-part IV model. For the first part, the elasticity equals -0.604 and is significant at a 5% level; a 10% increase in formal home-care hours would reduce the probability of receiving informal care by 6.04%. For the second part, the elasticity is -0.286 and is significant at a 10% level; a 10% increase in formal home-care hours would reduce by 2.86% the quantity of informal received by informal-care users. In Table 14, we calculate elasticities for both parts of our two-part IV model, when only personal formal home care is considered (the estimates of the model are presented in Table 12). For the first part, the elasticity equals -0.170 and is significant at a 5% level. A 10% increase in formal home-care hours would reduce the probability of receiving informal care by 1.70%. For the second part, the elasticity * : significant at the 10% level, * * : significant at the 5% level. We report the bias-corrected bootstrapped confidence intervals. * : significant at the 10% level, * * : significant at the 5% level. We report the bias-corrected bootstrapped confidence intervals.

is -0.076 and still significant at a 10% level. A 10% increase in formal home-care hours would reduce by 0.76% the quantity of informal received by informal-care users. Elasticities are lower than those calculated in Table 13, with total hours of formal home care. The crowding out of informal caregivers is weaker when only personal formal home care increases.

B Computation of elasticities

Elasticities computed in this article are average values. Elasticities are computed for each individual in our sample, and the mean of these elasticities is calculated. To obtain confidence intervals, we bootstrap this mean with 500 replicates. [START_REF] Cameron | Microeconometrics using stata[END_REF] [START_REF] Cameron | Microeconometrics using stata[END_REF] give the details of computing bootstrapped confidence intervals with Stata. Bias-corrected intervals are estimated following DiCiccio and Efron (1996) [START_REF] Diciccio | Bootstrap confidence intervals[END_REF].

B.1 Bivariate Tobit

For the bivariate Tobit model, the expectancy of y i conditional on being positive (the quantity informal care received for informal-care users) takes the following form:

E(y i |y i > 0, f c i , X i ) = exp(K + σ 2 2 ) Φ(σ + K σ ) Φ( K σ ) -1
with K = γ 0 + γ f c ln(1 + f c i ) + γ X X i and σ is the estimated standard error of the error term from the informal care equation. This expression differs from that expected for a Tobit model, as our variable of interest is ln(1 + y i ), which we derive from the first moment of a truncated log-normal distribution function (see [START_REF] William | Econometric Analysis[END_REF] [START_REF] William | Econometric Analysis[END_REF]).

We then have the expression of the expectancy of y i :

E(y i |f c i , X i ) = E(y i |y i > 0, f c i , X i ). Pr(y > 0)

E(y i |f c i , X i ) = exp(K + σ 2 2 )Φ(σ + K σ ) -Φ( K σ )
with K = γ 0 + γ f c ln(1 + f c i ) + γ X X i and σ is the estimated standard error of the error term from the informal-care equation.

The elasticity of y i (informal care) with respect to f c i (formal home care) can be written as follows:

i = ∂E(y i |f c i , X i ) ∂f c i . f c i E(y i |f c i , X i ) i = γ f c f c i 1 + f c i . exp(K + σ 2 2 )(Φ(σ + K σ ) + φ( K σ ) σ ) - φ( K σ ) σ exp(K + σ 2
2 )Φ(σ + K σ ) -Φ( K σ ) with K = γ 0 + γ f c ln(1 + f c i ) + γ X X i and σ is the estimated standard error of the error term from the informal care equation.

B.2 IV Probit

For the first part of the two-part model (the IV Probit part), the conditional expectancy of s i (the variable that equals 1 if the individual receives informal care) takes the following form: E(s i |f c i , X i ) = Pr(y i > 0|f c i , X i ) = Φ(γ 0 + γ f c ln(1 + f c i ) + γ X X i )

The elasticity of s i with respect to f c i (formal home care) can be written as follows:

i = ∂E(s i |f c i , X i ) ∂f c i . f c i E(s i |f c i , X i ) = γ f c f c i 1 + f c i . φ(γ 0 + γ f c ln(1 + f c i ) + γ X X i ) Φ(γ 0 + γ f c ln(1 + f c i ) + γ X X i )

B.3 IV Regress

For the second part of the two-part model (the IV Regress part), the expectancy of y i conditional on being positive (the quantity informal care received for informal-care users) takes the following form:

E(y i |y i > 0, f c i , X i ) = exp(β 0 + β f c ln(1 + f c i ) + β X X i + σ 2 2 )
with σ the standard error of the error term of the structural equation (informal care equation).

The elasticity of y i |y i > 0 with respect to f c i (formal home care) can be written as follows:

i = ∂E(y i |y i > 0, f c i , X i ) ∂f c i . f c i E(y i |y i > 0, f c i , X i ) = β f c f c i 1 + f c i

C Sensitivity analysis models

Figure 1 :

 1 Figure 1: Kernel density estimation of formal and informal care variables

Figure 2 :

 2 Figure 2: Kernel density estimation of logged formal and informal care variables

Table 1 :

 1 Summary statistics (means) for the entire sample and by kind of care used.

	Variable	All	All (Weighted)	Non-users	Formal only	Informal only	Both
		(N=1526)	(N=1526)	(N=224)	(N=657)	(N=250)	(N=395)
	Age	79.151 (8.461)	80.193 (8.060)	74.304 (8.621) 79.393 (7.896)	77.644 (8.047)	82.453 (8.030)
	Being a female	0.820 (0.384)	0.804 (0.397)	0.763 (0.426)	0.823 (0.382)	0.800 (0.401)	0.858 (0.349)
	Number of sons	1.117 (1.183)	1.154 (1.199)	1.040 (1.039)	0.992 (1.098)	1.280 (1.293)	1.263 (1.293)
	Number of daughters	1.109 (1.216)	1.055 (1.096)	0.955 (1.075)	0.901 (1.060)	1.424 (1.433)	1.342 (1.305)
	Dependence score	8.743 (8.252)	6.702 (6.814)	2.304 (3.055)	8.647 (7.682)	6.628 (6.059)	13.893 (9.153)
	Alzheimer	0.057 (0.232)	0.039 (0.195)	0.018 (0.133)	0.032 (0.177)	0.052 (0.222)	0.084 (0.277)
	Receiving formal care	0.689 (0.463)	0.663 (0.473)
	Receiving informal care	0.423 (0.494)	0.370 (0.483)
	Source: HSM 2008.
	Sample: elderly dependents aged 60 or older living alone in metropolitan France.
	Standard errors in parenthesis

Table 2 :

 2 Average quantity of care used by care users. HSM 2008. Sample: elderly dependents aged 60 or older living alone in metropolitan France, using formal and/or informal care. Standard errors in parenthesis

	Variable	All care users Formal only Informal only	Both
		(N=1302)	(N=657)	(N=250)	(N=395)
	Hours of formal care	8.522 (16.310)	9.694 (16.450)		11.966 (19.174)
	Hours of informal care	7.179 (15.377)		14.144 (19.701) 14.712 (19.022)

Source:

Table 3 :

 3 Spearman's correlation coefficient between formal and informal care variables

		Total sample	By level of dependence (Katz)
			Weak	Moderate	High	Very High
	ρ	-0.05 * *	-0.18 * * *	-0.30 * * *	-0.31 * * *	-0.30 * * *
	Source: HSM 2008.
	Sample: elderly dependents aged 60 or older living alone in metropolitan France.

* p < .1, * * p < .05, * * * p < .01

Table 4 :

 4 Bivariate Tobit model with formal home care

	Formal care	Informal care

* p < .1, * * p < .05, * * * p < .01

Table 5 :

 5 Bootstrapped elasticity

	Model	Bootstrap elasticity	A 10% increase in formal home-care hours
		[95% C.I.]	leads to a. . .
	Bivariate Tobit model
	E(y)	-0.425 * * [-0.646, -0.167] 4.25% decrease in the quantity of
			care received.

Source: HSM 2008. Sample: elderly dependents aged 60 or older living alone in metropolitan France.

Table 6 :

 6 Bivariate Tobit model with personal formal home care * p < .05, * * * p < .01

		Personal formal home care	Informal care
	Hours of personal formal care		-0.789 * * * (0.302)
	Dependence score	0.154 * * * (0.00946)	0.117 * * * (0.0154)
	Alzheimer (Ref: No)	1.230 * * * (0.408)	1.178 * * * (0.258)
	Has a diploma (Ref: No)	-0.0334 (0.148)	-0.162 (0.129)
	Income (Ref: Less than 600 e)
	600 e-1000 e	-0.195 (0.218)	-0.0128 (0.150)
	1000 e-1500 e	0.0829 (0.226)	-0.0249 (0.206)
	1500 e and +	-0.0424 (0.324)	-0.910 * * * (0.287)
	Missing	0.367 (0.312)	-0.234 (0.242)
	Number of sons	0.0423 (0.0616)	0.133 * * * (0.0475)
	Number of daughters	-0.104 (0.0737)	0.360 * * * (0.0605)
	Is a female (Ref: No)	0.259 (0.245)	-0.340 * (0.174)
	Age group (Ref: 60-64)
	65-69	0.734 (0.458)	0.681 * * (0.288)
	70-74	0.918 * * (0.427)	0.441 * (0.261)
	75-79	1.093 * * * (0.373)	0.546 * * (0.268)
	80-84	1.221 * * * (0.439)	0.799 * * * (0.304)
	85-89	1.662 * * * (0.405)	1.116 * * * (0.273)
	90+	1.546 * * * (0.519)	1.557 * * * (0.355)
	Density of midwives	0.211 * * * (0.0318)
	Intercept	-5.746 * * * (0.439)	-1.873 * * * (0.342)
	σ	2.304 * * * (0.117)	2.320 * * * (0.0874)
	ρ	0.302 * * (0.144)
	Observations	1526
	Standard errors in parenthesis

* p < .1, *

Table 7 :

 7 Bootstrapped elasticities for personal formal home care only

	Model	Bootstrap elasticity	A 10% increase in
		[95% C.I.] for personal personal formal home-care hours
		formal home care	leads to a. . .
	Bivariate Tobit model
	E(y)	-0.128 * * [-0.204, -0.057]	1.28% decrease in the quantity of informal
			care received.
	Source: HSM 2008.
	Sample: elderly dependents aged 60 or older living alone in metropolitan France.

Table 8 :

 8 Bootstrapped elasticities with the sunshine instrumental variable (sensitivity analysis)

	Model	Bootstrap elasticity	Bootstrap elasticity
		[95% C.I.] for total	[95% C.I.] for personal
		formal home care	formal home care
	Bivariate Tobit model
	with the sunshine instrument
	(sensitivity analysis)
	E(y)	-0.399 * * [-0.659,-0.130] -0.120 * * [-0.194,-0.055]

Source: HSM 2008. Sample: elderly dependents aged 60 or older living alone in metropolitan France.

Table 9 :

 9 Bootstrapped elasticities with the Katz dependence score (sensitivity analysis)

	Model	Bootstrap elasticity	Bootstrap elasticity
		[95% C.I.] for total	[95% C.I.] for personal
		formal home care	formal home care
	Bivariate Tobit model
	with the Katz dependence score
	(sensitivity analysis)
	E(y)	-0.270 * * [-0.514,-0.034] -0.074 * * [-0.148,-0.001]

Source: HSM 2008. Sample: elderly dependents aged 60 or older living alone in metropolitan France.

Table 10 :

 10 Two-part IV model

	IV Probit: Receive Inf. Care	IV Regress (2SLS) : Hours of Inf. Care
	Instrumental eq.	Structural eq.	Instrumental eq.	Structural eq.

Table 11 :

 11 Two-part model with formal home care considered as exogenous

		Probit: Receive Inf. Care	OLS: Hours of Inf. Care
	Hours of formal care	-0.438 * * * (0.0495)	-0.142 * * * (0.0525)
	Dependence score	0.0718 * * * (0.00646)	0.0553 * * * (0.00670)
	Alzheimer (Ref: No)	0.486 * * * (0.174)	0.557 * * * (0.160)
	Has a diploma (Ref: No)	0.0242 (0.0644)	-0.134 (0.0824)
	Income (Ref: Less than 600e)
	600e-1000e	0.000950 (0.0760)	0.0610 (0.0970)
	1000e-1500e	-0.0524 (0.0986)	0.0341 (0.136)
	and +	-0.401 * * * (0.150)	-0.253 (0.199)
	Missing	-0.148 (0.118)	-0.104 (0.192)
	Number of sons	0.0563 * * (0.0270)	0.0419 (0.0268)
	Number of daughters	0.175 * * * (0.0335)	0.111 * * * (0.0419)
	Is a female (Ref: No)	-0.0784 (0.0888)	-0.455 * * * (0.116)
	Age group (Ref: 60-64)
	65-69	0.331 * * (0.138)	0.286 (0.329)
	70-74	0.301 * * (0.119)	0.238 (0.200)
	75-79	0.412 * * * (0.128)	0.333 (0.257)
	80-84	0.557 * * * (0.147)	0.428 * (0.239)
	85-90	0.584 * * * (0.142)	0.618 * * (0.246)
	90+	0.886 * * * (0.175)	0.689 * * * (0.231)
	Intercept	-0.873 * * * (0.150)	1.363 * * * (0.292)
	Observations	1526	637
	Standard errors in parenthesis

* p < .1, * * p < .05, * * * p < .01

Table 12 :

 12 Two-part IV model with personal formal home care

	IV Probit: Receive Inf. Care	IV Regress (2SLS) : Hours of Inf. Care
	Instrumental eq.	Structural eq.	Instrumental eq.	Structural eq.

Table 13 :

 13 Bootstrapped elasticities (IV two-part model) HSM 2008. Sample: elderly dependents aged 60 or older living alone in metropolitan France.

	Model	Bootstrap elasticity	A 10% increase in formal home-care hours
		[95% C.I.]	leads to a. . .
	Two-Part model
	Pr(y > 0)	-0.604 * * [-0.858, -0.112] 6.04% decrease in the probability of informal
			care use.
	E(y|y > 0)	-0.286 * [-0.967, 0.038]	2.86% decrease in the quantity of informal
			care received for informal-care users.

Source:

Table 14 :

 14 Bootstrapped elasticities (IV two-part model) for personal formal home care only

	Model	Bootstrap elasticity	A 10% increase in
		[95% C.I.] for personal personal formal home-care hours
		formal home care	leads to a. . .
	Two-Part model
	Pr(y > 0)	-0.170 * * [-0.295, -0.053]	1.70% decrease in the of informal
			care use.
	E(y|y > 0)	-0.076 * [-0.153, 0.018]	0.76% decrease in the quantity of informal
			care received for informal-care users.

Source: HSM 2008. Sample: elderly dependents aged 60 or older living alone in metropolitan France.

Table 15 :

 15 Bivariate Tobit model with the sunshine instrumental variable (sensitivity analysis) p < .1, * * p < .05, * * * p < .01

	Formal home care	Informal care

*

Table 16 :

 16 Bivariate Tobit model with the Katz index of dependence (sensitivity analysis) p < .1, * * p < .05, * * * p < .01

	Formal home care	Informal care

*

We use the well-known Katz index of dependence, an index often used to evaluate the level of dependence of elderly dependents in the United States. The original index has eight modalities, but we group them into four, with each catching enough individuals for statistical significance: "being weakly dependent", "being moderately dependent", "being highly dependent", and "being very highly dependent".

Acknowledgements

We gratefully acknowledge financial support from the Université Paris-Dauphine. We thank professors from the LEDa-LEGOS for their helpful comments during seminars. We are especially grateful to Jérôme Wittwer, Eve Caroli, Roméo Fontaine, Eric Delattre, Céline Pilorge, and participants to the JMA 2012, the LEDa-LEGOS seminars, the JESF 2012 for useful comments and suggestions; the authors alone are responsible for the analysis and interpretation in this paper.

Appendices A A two-part model with an instrumental variable A.1 Description of the IV two-part model

The two-part model was introduced by [START_REF] Duan | A comparison of alternative models for the demand for medical care[END_REF] [START_REF] Duan | A comparison of alternative models for the demand for medical care[END_REF]. It allows the separation of behavior into two stages: a Probit model that predicts the probability of receiving any informal care and a least-squares model to predict the continuous amount of informal care received when informal care is given. The use of informal care (y) is a function of formal home care (f c) and of a vector of exogenous characteristics of the individual (X). The subscript i represents the individual. The two-part model assumes that part one, Pr(y i > 0), is described by a Probit model such that:

where Φ(.) represents the cumulative density function of the standard normal distribution, γ 0 and γ f c are the parameters to be estimated and γ X is a vector of parameters to be estimated. Part two corresponds to the following equation, assuming that the logarithm of the positive values of y is linear in ln(1 + f c) and X:

where β 0 , β f c and β X are the parameters (or vectors of parameters) to be estimated.

Because the formal home-care variable (f c) is skewed, we take the log in both parts of our model. The log of the dependent variable y in the second part is taken to diminish the influence of outliers. To treat the problem of endogeneity of formal home care, we estimate instrumental-variable (IV) models that produce consistent parameter estimates. For the first part of our two-part model, the discrete outcome of utilization, we use the instrumental variables Probit (IV Probit) ado program in Stata 11, estimated with maximum-likelihood techniques. Part two (continuous utilization) uses the standard two-stage least-squares estimation.

This two-part IV model could be well suited to the question (cf. Van Houtven and Norton (2004) [START_REF] Van Houtven | Informal care and health care use of older adults[END_REF] and [START_REF] Bonsang | Does informal care from children to their elderly parents substitute for formal care in Europe?[END_REF] [START_REF] Bonsang | Does informal care from children to their elderly parents substitute for formal care in Europe?[END_REF]), but it does not account for censoring of the endogenous variable of formal home care. In contrast to Heckman's selection model, both parts of the two-part model are independent, and the probability of receiving informal care and the amount of informal care received are independent, which is a very strong assumption. Nevertheless, it allows the decision to receive informal care to be modeled separately from the quantity received when informal care is given.

A.2 Estimates of the IV two-part model

The estimates of the two-part IV model are presented in Table 10. We first examine the instrumental equations of the IV Probit and the IV Regress, explaining the quantity of formal home care received. Our instrument is the density of self-employed midwives in each French district. The variable is significant at the 1% level and is thus a strong predictor of formal home care, considering both our entire sample and the sample of informal-care users only. The adjusted-R 2 values of our two equations are 0.42 and 0.43, respectively; the explanatory power of our instrumental equations is thus very high, so formal home care is not replaced by a noisy measure in IV estimation. Notably, the two main factors seem to explain the quantity of formal home care received, i.e., the level of dependence (the dependence score, age group and Alzheimer's disease are strong predictors of formal home-care utilization) and the education level (a diploma is a positive predictor of the use of formal home care).