5 A recent paper by Courbage and Zweifel (2011) suggests to look at intra-family moral

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2 Introduction Long-term care has been increasingly becoming a "hot" toic over the last years, and its imortance is redicted to grow even more in the coming decades. Long-term care is the care for eole who are deendent on the hel of others in their basic daily activities (such as dressing, bathing, eating, etc.). It can be seen as consisting of both health and social care which can be rovided both formally (by aid rofessional caregivers) and informally (by family members or friends), at home and in secial institutions. The need for long-term care is highly related with age. For instance, around half of all long-term care users in the OEC countries are over 80 years old (OEC, 2011). Therefore, more and more attention to long-term care issues is required due to the aarent ageing of the oulations in most industrialised countries. Indeed, it is exected that in the OEC the share of the oulation aged 80 and over will increase from 4% in 2010 to 9.4% in 2050 (com ared to 1% in 1950) (OEC, 2011). The Euroean Union has estimated otential increases in the number of deendent old ersons under different scenarios and redicts that, even assuming a decrease in age-secific disability rates in the future, the number of deendent old eole in the EU27 will rise by 90% from 2007 to 2060 (Euroean Commission, 2009). If the disability rates by age remain constant, the redicted increase is even more drastic and reaches 115%, which means that the number of deendent elderly is exected to more than double (Euroean Commission, 2009). In the light of these trends, it is imortant to make sure that our societies are ready to successfully face the challenge. For the moment, however, a number of unresolved issues exist, which calls for devoting considerable attention to them. Long-term care is currently rovided and financed by three institutions: the family, the State and the market (Cremer et al., 2012). Most care is rovided by family caregivers. Even though it is difficult to obtain recise data on informal caregiving, and the numbers vary across countries, family carers considerably outnumber formal caregivers in most countries, and in some cases their number is estimated to be more than ten times larger than that of formal care roviders (for examle, in the Netherlands, Canada, United States, New Zealand) (OEC, 2011). However, it is not clear whether the current situation will last. The societal changes such as the rising number of childless families, the increasing female articiation in the labour market, the mobility of children, higher divorce rates or changing family values are likely to decrease the availability of family hel (Cremer et al., 2012; OEC, 2011). Therefore, with the drastically rising demand for long-term care, it is becoming increasingly imortant to ensure reliable alternatives to family solidarity. State intervention can be seen as one of the solutions. In most industrialised countries, the governments articiate in the rovision and/or financing of longterm care services. However, the extent and the nature of this articiation is rather different across countries. In addition to this, very few countries have exlicit social long-term care insurance schemes (Cremer et al., 2012; OEC, 2

3 2011). In the resence of the current demograhic and societal trends, there is thus a need to assess the suitability of the current olicy designs and to look for the best ways of ublic intervention, which among other things should take into account its ossible effects on the family and the market. Turning to the market, another source of hel in the roblem of deendence could be rivate long-term care insurance. However, here we are facing another issue, namely, the so-called long-term care insurance uzzle. This uzzle has been created by a surrisingly low demand for rivate long-term care insurance. In articular, even though the otential long-term care costs are large 1 and the robability to become deendent is high, 2 only a small fraction of individuals urchase rivate long-term care insurance. A number of different factors otentially exlaining this uzzle have been roosed in the literature. Those ossible exlanations range from the ones that assume erfect rationality of individuals (for instance, the high rice of insurance or crowding out by the State) to the ones considering the non-urchase of insurance as irrational (such as myoia or ignorance and denial of the issue of deendence). 3 Even though it seems that none of the roosed reasons alone is able to entirely exlain the uzzle, each of them might be laying some role in the issue. Therefore, being able to solve or at least reduce (some of) these otential roblems might hel to stimulate the market for rivate long-term care insurance. This aer is concerned with one of these otential roblems. In articular, it is the roblem of intra-family moral hazard, which was roosed as a ossible exlanation for the long-term care insurance uzzle by Pauly (1990). The idea of the intra-family moral hazard argument is the following. In the case of deendence, exenditures on formal long-term care reduce the deendent erson's wealth (and thus the bequest that will be left to his/her children). The children can hel to avoid these formal care exenses (and thus rotect the future bequest) by roviding care to their arent themselves. However, if the arent has long-term care insurance, the cost of formal care is (at least artly) covered by the insurer. Thus, long-term care insurance also rotects the arent's wealth (and the future bequest), 4 which results in the children having less incentive to rovide care themselves. Therefore, a arent who refers being taken care of by his/her children rather than by unknown formal caregivers might be discouraged from buying long-term care insurance. While the argument is develoed with little formalization in Pauly (1990), Zweifel and Striiwe (1998) rovide a more rigorous model for this idea and show that it is indeed reasonable to believe that the intra-family moral hazard effect might be a cause of the non-urchase 1 For instance, a nursing home stay in the U.S. costs between $ and $ er year, while the average cost in France is around er year (Taleyson, 2003). 2 According to Kemer and Murtaugh (1991), a 65-year-old erson has a robability of 43% to enter a nursing home at some time before his/her death (this robability is 52% for women and 33% for men). See also Norton (2000). 3 For recent surveys of otential exlanations for the uzzle, see Cremer et al. (2012), Pestieau and Ponthiere (2011) and Brown and Finkelstein (2011). 4 Pauly (1990, 1996) names the rotection of one's bequest as the main function of long-term care insurance. 3

4 of long-term care insurance. 5 However, the analysis of Zweifel and Struwe (1998) only focuses on one tye of insurance benefits, namely, benefits which are roortional to the amount of long-term care exenditures. The resent aer uts forward the idea that the extent of the above described roblem might be different deending on the form of insurance benefits. The roortional benefits analyzed by Zweifel and Struwe (1998) roughly cature the idea behind long-term care insurance benefits revalent in the U.S where these benefits take the form of a reimbursement for the long-term care exenditures the insured has, in the limit of a certain reimbursement ceiling (uran and Taleyson, 2003). However, this is not the only tye of long-term care insurance benefits used in ractice. For instance, rivate insurers in France use a substantially different benefit scheme. In articular, insurance ayments in France are of the form of a fixed monthly cash benefit which starts being aid if the insured is recognized to be deendent. This benefit does not deend on the amount of long-term care exenditures the erson has and can be used by the erson in the way he/she decides. This clearly contrasts with the U.S. where, in order to be reimbursed, the long-term care exenditures have to come from the use of services that are aroved by the insurer (uran and Taleyson, 2003). Thus, in the American scheme the insured can only benefit from the insurance if he/she uses certain services of formal care. This is not the case with fixed insurance benefits. Therefore, intuitively it seems that the use of formal care should be encouraged much less in the case of fixed insurance benefits, which should hel to limit intra-family moral hazard. Some emirical studies reviewed below seem to suggest this idea as well. If the effect of intra-family moral hazard is strong enough, it seems likely that the fear to distort the children's incentives might offset altruistic considerations (such as intentions to ensure sufficiently large bequests to their children) that arents could have in their insurance urchase decisions. Nevertheless, interestingly enough, this does not seem to be the case in France. In articular, a study conducted by Courbage and Roudaut (2008) finds emirical evidence that in France long-term care insurance urchases are driven by altruistic motives. First, they find that the demand for long-term care insurance rises with the robability of leaving a bequest, which confirms Pauly's (1990, 1996) argument that long-term care insurance is urchased to rotect one's bequest. Further, they find that long-term care insurance is ositively associated with the number of children and the fact of living with a artner. Finally, it is found that long- 5 A recent aer by Courbage and Zweifel (2011) suggests to look at intra-family moral hazard as at a two-sided henomenon arguing that aart from the just described effect of insurance on the incentives for children (one side of the henomenon), the second side exists in that arents may urchase less insurance if they can rely on the caregiving effort of their children. However, the caregiving effort in their aer is modeled as a reventive one that hels to kee the arent out of a nursing home (which more generally can be seen as actually reventing from a need for long-term care). In the resent aer, I consider the children's caregiving as the rovision of care when the arents' need for long-term care has already materialized and focus on the first side of the henomenon (the incentives for children), as suggested by Pauly (1990) and Zweifel and Striiwe (1998). 4

5 term care insurance is ositively associated with the robability of receiving informal care in the case of such a need in the future. These findings suggest that eole are not afraid that insurance coverage might distort caregiving incentives of their relatives. In other words, the findings seem to suggest that in France the degree of intra-family moral hazard is likely to be rather low. For comarison, one could consider the study made by Sloan and Norton (1997) who do not find evidence for altruistic bequest motive in long-term care insurance decisions in the U.S. They do not detect any relationshi between stated altruism (i.e. the erson's wish to leave a bequest) and demand for longterm care insurance. This suggests that a arent might well be willing to leave a bequest, but still decide not to urchase long-term care insurance. While it is ossible that some other reasons exist, one exlanation for this could be the attemt to avoid intra-family moral hazard. Therefore, one could susect that intra-family moral hazard is a more serious roblem in the U.S. than in France. While it is ossible that some other factors come into lay as well, a otential exlanation for this difference could be the different nature of insurance benefits. The aim of this aer is to exlore more formally the roles of fixed and roortional insurance benefits in the issue of intra-family moral hazard. In articular, in the context of intra-family moral hazard, the aer comares the imact of fixed and of roortional insurance benefits on the long-term care related behavior of children and eventually on the insurance urchase decisions of arents. The model considers an elderly arent and his adult child. The arent might become deendent, in which case he laces a secial value on the long-term care received from his child. The arent also cares about the bequest that will be left to the child; thus, he might want to urchase insurance in order to rotect his bequest from long-term care exenses. If the arent becomes deendent, the child chooses the amount of care she wants to rovide. I consider searately the cases when the child likes and dislikes roviding care to the arent. In addition to the amount of caregiving, the child's utility deends on her wealth which consists of her earnings in the labour market and the bequest left by the arent. The bequest is the channel through which the child's care rovision is affected by the insurance coverage urchased by the arent before the materialization of the risk of deendence. Reasoning backwards, the first art of the analysis studies the roblem of the child and comares the effects that insurance with fixed and roortional benefits has on the child's choice of caregiving. The second art of the analysis exlores how these different effects (anticiated by the arent) influence the arent's insurance urchase decisions. The model is similar to the one of Zweifel and Striiwe (1998). However, as mentioned above, Zweifel and Striiwe (1998) analyze only the case of roortional benefits and thus do not make any comarisons. Also, the model in this aer differs from theirs in a number of other asects such as, for instance, the way the roortional benefits are modeled. In addition, Zweifel and Striiwe (1998) model the arent and the child as the rincial and the agent. I do not adot this aroach here but rather follow the idea of equal treatment of both 5

6 arties suggested by Courbage and Zweifel (2011). 6 It should be also mentioned that, differently from Zweifel and Striiwe (1998), the resent aer looks at the roblem of the child from the ersective of the standard consumer's theory, which allows to identify and understand the substitution and income effects that influence the child's choice. The aer finds that the amount of caregiving of a child who dislikes roviding care is decreasing in the arent's insurance coverage both in the case of roortional and in the case of fixed insurance benefits. Thus, intra-family moral hazard exists with both tyes of benefits. However, it is also shown that the same amount of long-term care insurance urchased by the arent results in more care given by the child when insurance benefits are fixed than when they are roortional. This confirms that fixed insurance benefits indeed hel to mitigate the henomenon of intra-family moral hazard. On the other hand, intra-family moral hazard is never a roblem with fixed insurance benefits when the child likes roviding care. In that case, fixed benefits not only eliminate intra-family moral hazard but also trigger an oosite effect, namely, an increase in the child's caregiving. With roortional benefits, both an increase and a decrease in the child's care rovision is ossible, which means that intrafamily moral hazard cannot be comletely ruled out. In addition, even in the case when intra-family moral hazard is not a roblem with either tye of benefits, it is shown that the child's chosen amount of caregiving is greater with fixed than with roortional insurance benefits. It is also interesting to note that the child's utility is higher with fixed rather than with roortional benefits. Turning to the arent, it is shown that, in the context of intra-family moral hazard, fixed insurance benefits make insurance more desirable to him. Even though fixed benefits do not always guarantee that the arent will find insurance beneficial, urchasing a ositive amount of insurance is more likely to be in the arent's interest when benefits are fixed rather than roortional. The results of the aer indeed confirm that a low degree of intra-family moral hazard in France could (at least artly) be attributed to the use of fixed insurance benefits. Moreover, the results could also artly exlain the relative success of the French rivate long-term care insurance market. While the French and the American markets are the world's two most develoed markets for rivate long-term care insurance, in 2010, about 15% of the oulation aged 40 and over had rivate long-term care insurance in France, comared to only 5% in the U.S. (OEC, 2011). Given the findings of the aer, it seems reasonable to believe that the use of fixed benefits takes art in the success of France. Indeed, if fixed benefits (through the softening of intra-family moral hazard) can make it more likely that a arent will decide to urchase insurance, the use of this insurance scheme could be seen as a reason for the success of attracting customers in the French long-term care insurance market. 6 However, in Courbage and Zweifel (2011) the arent's choice of insurance and the child's choice of care rovision are modeled as simultaneous decisions. Here the choices are sequential since the arent has to make the insurance decision first (before one of the two ossible states of nature materializes), and the child chooses the amount of care only when (and if) the state of nature with the need for long-term care materializes. 6

7 1 π c π EU = πu +(1 π)u I U = u (Q, W ) U I = v (X I,W I )

8 X I W I L L, L, + L (Q Q / >0 W δ γ [0, 1] L Q W πγδ( L Q) W = W πγδ( L Q) (1 γ)δ( L Q) W I = W πγδ( L Q) X I L Q

9 γ γ B πb W = W πb δ( L Q)+B B W I = W πb X I B U c = u c (Q, W c ) Q (W c ) c > 0 c < 0 W c = w( L Q)+W w ( L Q) L

10 Q πγδ( L Q) Q. πγδ( L Q(γ)) Q(γ) Q γ Q γ Q(γ) c > 0 c < 0 c < 0 c < 0 γ B w w δ

11 (1 γ)δ w<δ Q =0 Q w w<(1 γ)δ w max U c Q,Wc = u c (Q, Wc ) s.t. W c = w( L Q)+ W πγδ( L Q) (1 γ)δ( L Q) W c W c = W πγδ( L Q) [(1 γ)δ w]( L Q) [(1 γ)δ w]q = W πγδ( L Q) [(1 γ)δ w] L Q W c (1 γ)δ >w [(1 γ)δ w] r W c r Q = W πγδ( L Q) r L I r

12 πγδ( L Q) I r f = δ w > r max U c Q,Wc = u c (Q, Wc ) s.t. W c f Q = W πb + B f L I f Q r I r f I f Q = Q + Q i i I i i = r, f Q i i L Q c < 0 I i < 0 Q. ( r,i r ) γ = r r γ + I r I r γ = δ Q + Q + r I r + πδ( L Q)+πγδ I r γ + δ L

13 ( ) δ Q γ = r + γ Q I r ( δ)+ 1 I r πγδ γ ( ) I r [δ L(1 π)+πδq] I r γ = ( ) δ Q r + ( ) I r δ(1 π)( L Q) 1 I r πγδ < 0 ( f,i f ) B = f f B =0 + I f I f B = (1 π) < 0 I f ( )

14 w<(1 γ)δ B d[γδ( L Q)] = 1 δ( L Q) γδ dγ =1 γ dγ = 1 δ( L Q) γδ γ B = (1 π) I f γ dγ d[γδ( L Q)]=1 = δ Q r + I r δ(1 π)( L Q) 1 I r πγδ 1 δ( L Q) γδ γ γ dγ d[γδ( L Q)]=1 B

15 γ dγ d[γδ( L Q)]=1 γ=0 = I r γ=0 = I f B=0 1 ( L Q) B B=0 = B=0 (1 π) I f Q γ=0 + γ=0 (1 π) r I r w<(1 γ)δ w<(1 γ)δ c < 0) L Q)

16 [(1 γ)δ w] ( L Q) r πγδ( L Q) r πγδ( L Q) r [δ w] γδ( L Q) r B B = γδ( L Q) r [δ w] ( L Q) f < ( L Q) r

17 c > 0 w w (1 γ)δ Q = L) (1 γ)δ <w δ Q = L w>δ w max U c Q,Wc = u c (Q, Wc ) s.t. W c = w( L Q)+ W πγδ( L Q) (1 γ)δ( L Q) W c W c = W πγδ( L Q)+[w (1 γ)δ]( L Q) +[w (1 γ)δ]q = W πγδ( L Q)+[w (1 γ)δ] L w>(1 γ)δ [w (1 γ)δ]

18 ρ r W c + ρ r Q = W πγδ( L Q)+ρ r L M r M r ρ f = w δ < ρ r max U c Q,Wc = u c (Q, Wc ) s.t. W c + ρ f Q = W πb + B + ρ f L M f ρ f M f Q ρ r M r ρ i Q = Q Q ρ i ρ i M i M i > 0 i = r, f Q. Q ρ i (ρ r,m r ) γ = ρ r ρ r γ + M r M r γ + πδ( L Q)+πγδ M r γ + δ L = δ Q Q + ρ r M r

19 γ = ( ) ρ r ( ) δ Q Q δ + M r γ M r [δ L(1 π)+πδq] 1 M r πγδ (?) M r γ = ( ) δ Q ρ r + M r δ(1 π)( L Q) 1 M r πγδ (?) 0 (ρ f,m f ) B = ρ f ρ f B =0 + M f M f B = (1 π) > 0 M f

20 w>δ γ dγ d[γδ( L Q)]=1 γ=0 = 1 ( L Q) Q ρ r γ=0 ( ) + γ=0 (1 π) M r B B=0 = B=0 (1 π) M f 1 ( L Q) Q γ=0 > γ=0 (1 π), ρ r M r ( )

21 1 ( L Q) Q ρ r γ=0 ( ) = γ=0 (1 π) M r 1 ( L Q) Q γ=0 < γ=0 (1 π), ρ r M r ( ) M r γ=0 = M f B=0 w>δ w>δ [w (1 γ)δ] Q r πγδ( L Q r )

22 πγδ( L Q r ) [w δ] γδ( L Q r ) B B = γδ( L Q r ) [w δ] Q f >Q r

23 only eliminate this henomenon but also trigger an oosite effect, namely, an increase in the child's caregiving. It is true that an increase in the child's care rovision is ossible with roortional benefits as well; however, in any event, the child's chosen amount of caregiving is always greater with fixed insurance benefits. Therefore, even in the case when intra-family moral hazard is not a roblem, fixed insurance benefits still allow the arent to enjoy more care given by his child. Summing u the results of this section, it first should be noted that the imact of insurance on the child's caregiving is quite different in the two cases of the child's references analyzed. In the case when the child dislikes roviding care (and has a low wage), insurance unambiguously decreases the amount of her caregiving with both tyes of insurance benefits. In the case when the child likes roviding care (and has a high wage), with fixed insurance benefits, the effect is always oosite: insurance increases the child's caregiving. With roortional benefits, the effect of insurance cannot be unambiguously determined. However, irresective of whether we look at Case 1 or Case 2, it is clear that, when the amount of insurance is the same with both tyes of benefits, the child's care rovision is greater when insurance benefits are fixed. It is also interesting to note that, again indeendently of the case analyzed, the child achieves more utility with fixed rather than with roortional insurance benefits. We now turn to the analysis of the arent. 3. Fixed vs Proortional: The imact on the arent's insurance decision Having comared the imact of insurance on the child's care rovision in the cases of fixed and roortional benefits, we can now see how this affects the arent's insurance urchase decisions. In this section, I first comare the arent's exected utility with fixed and with roortional insurance benefits and then analyze in more detail the arent's decision of whether to buy long-term care insurance or not. As in the revious section, I discuss searately the cases when the child likes and dislikes roviding care Parent's exected utility This subsection looks at the arent's exected utility with the two tyes of insurance benefits. Let us start with the case when the child dislikes roviding care. 23

24 c < 0 EU = πu +(1 π)u I U = u (Q, W ) U I = v (X I,W I ) γ w<(1 γ)δ γ w<(1 γ)δ Q r γδ( L Q r ) B γδ( L Q r ) W Q f >Q r Q f B = γδ( L Q r ) Q W = W πγδ( L Q r ) (1 γ)δ( L Q r )= = W πγδ( L Q r ) =πb δ( L Q r )+γδ( L Q r ) =B W = W πb δ( L Q f )+B Q f >Q r Q W

25 W I = W πγδ( L Q r ) X I =πb W I = W πb X I (1 γ)δ w<δ w<δ w δ c > 0 w>δ

26 w>δ c < 0 maxeu = πu (Q, W γ,x I )+(1 π)v (X I,W I ) γ X I X I

27 v X I v X I + v I I X I γ v W I =0 =0 EU γ = π γ + W γ v +(1 π) X I X I γ + v I I = γ = π γ + δ( L Q)(1 π) γδ (1 π)+δ γ γ +(1 π) πγδ v γ πδ( L Q) I W + maxeu = πu (Q, W B, X I )+(1 π)v (X I,W I ) B X I B EU B = π B + W B v +(1 π) X I = π B + (1 π)+δ B +(1 π)[ π] v W I X I B + v I I = B +

28 EU γ EU B EU γ γ=0 = π(1 π)δ( L Q) γ=0 v I γ=0 +π γ γ=0 γ=0 + δ W γ=0 ( ) EU B B=0 = π(1 π) B=0 v I B=0 +π B B=0 ( ) B=0 B=0 + δ w<(1 γ)δ γ γ =0 w<δ Q Q Q W γ=0 = W B=0 v I γ=0 = v I B=0 > 0 Q Q Q v (X I,W I )=lnx I + lnw I lnw c Q W > 2δ(δ w) δ+w (Q, W ) W = 1 W u c (Q, W c )=

29 w < (1 γ)δ EU γ dγ d[γδ( L Q)]=1 +π γ=0 = π(1 π) γ=0 v I γ=0 + γ dγ d[γδ( L Q)]=1 γ=0 ( ) γ=0 γ=0 + δ EU B B=0 = π(1 π) B=0 v I B=0 +π B B=0 ( ) B=0 B=0 + δ W γ=0 v I γ=0 = B=0 v I B=0 γ=0 + δ W γ=0 = B=0 + δ W B=0

30 γ w<(1 γ)δ EU B B=0 0 EU γ γ=0 < 0 EU γ γ=0 0 EU B B=0 > 0

31 c > 0 w>δ EU γ γ=0 EU B B=0 EU γ γ=0 = π(1 π)δ( L Q) γ=0 v I γ=0 +π γ γ=0 γ=0 + δ W γ=0 /( ) EU B B=0 = π(1 π) B=0 v I B=0 +π B B=0 B=0 B=0 + δ EU B B=0 Q γ γ=0 0 EU γ γ=0 γ γ=0 < 0

32 w>δ

33 might have a somewhat different imact on the behavior of children and arents than benefits roortional to long-term care exenditures do. In articular, the aer has studied and comared the effects that insurance with fixed and roortional benefits has on the caregiving choice of the child and, consequently, the influence that these different effects have on the insurance urchase decisions of the arent who values the care rovided by his child. The results of the aer formally confirm the intuitive hyothesis that fixed insurance benefits should hel to limit the henomenon of intra-family moral hazard. Indeed, in the case of a child who dislikes roviding care, the aer finds that, even though the child's caregiving is decreasing in the insurance coverage with both tyes of benefits, the same amount of long-term care insurance urchased by the arent results in more care rovided by the child when insurance benefits are fixed than when they are roortional. Thus, with fixed benefits the child reduces her caregiving less drastically. The amount of the child's care rovision is greater with fixed than with roortional benefits in the case when she likes roviding care as well. In that case, fixed benefits even eliminate intrafamily moral hazard comletely, whereas the effect of roortional benefits is ambiguous. The fact that the child rovides more care when benefits are fixed rather than roortional makes the arent refer insurance with fixed benefits: the same amount of insurance gives him a higher exected utility when benefits are fixed. However, the arent will not always be willing to buy a ositive amount of insurance even when fixed benefits are available; nevertheless, it is shown that urchasing a ositive amount of insurance is more likely to be in the arent's interest when benefits are fixed rather than roortional. In other words, the non-urchase of long-term care insurance is more likely with roortional benefit schemes, which could artly exlain why rivate long-term care insurance is urchased by a larger share of the targeted oulation in France than in the U.S. Besides adding to the exlanation of the success of the French insurance scheme, the findings of the aer can be of some use to ublic olicy as well. Zweifel and Striiwe (1998) have raised concerns about the welfare effects that are/could be caused by a comulsory social long-term care insurance. In articular, since (because of intra-family moral hazard) arents might in many cases find rivate long-term care insurance undesirable, imosing a comulsory social insurance might go against the interests of many arents. The analysis in this aer has shown that insurance can be made more desirable (or less undesirable) to arents by using fixed benefit schemes. Indeed, it is shown that insurance is more likely to be in the interest of the arent when benefits are fixed rather than roortional. Also, even in the cases when fixed benefits do not hel to make insurance desirable to the arent, his exected utility is still higher with fixed than with roortional benefits. In addition to this, the aer also finds that the child is better-off with fixed benefits as well. Therefore, based on these results, it seems reasonable to favour ublic schemes with fixed rather than roortional benefits. 33

34 Finally, it should be reminded that, as mentioned in the introduction, the analysis in this aer is focused on the context of intra-family moral hazard and thus abstracts from other factors and situations that could otentially modify the conclusions about the desirability of fixed and roortional insurance benefits. Thus, rather than drawing general inferences, the aer rovides conclusions based on a articular context which, nevertheless, constitutes a otential iece of the long-term care insurance uzzle and is therefore imortant to understand. References [1] Brown, Jeffrey R. and Amy Finkelstein. "Insuring Long-Term Care in the United States". Journal of Economic Persectives, Vol. 25, No. 4 (2011), [2] Courbage, Christohe and Nolwenn Roudaut. "Emirical Evidence on Long-Term Care Insurance Purchase in France". The Geneva Paers on Risk and Insurance- Issues and Practice, Vol. 33 (2008), [3] Courbage, Christohe and Peter Zweifel. "Two-Sided Intergenerational Moral Hazard, Long-Term Care Insurance, and Nursing Home Use". Journal of Risk and Uncertainty, Vol. 43 (2011), [4] Cremer, Helmuth, Pierre Pestieau and Gregory Ponthiere. "The Economics of Long-Term Care: A Survey". CORE iscussion Paer (2012). [5] Cutler, avid. "Why oesn't the Market Fully Insure Long-Term Care?''. NBER Working Paer No (1993). [6] uran, Romain and Lucie Taleyson. "Les Raisons du Succes de l' Assurance eendance en France". Risques - Les Cahiers de l' Assurance, Vol. 55 (2003), [7] Euroean Commission. "The 2009 Ageing Reort: Economic and Budgetary Projections for the EU-27 Member States ( )". Joint Reort reared by the Euroean Commission (G ECFIN) and the Economic Policy Committee (AWG), [8] Kemer, Peter and Christoher M. Murtaugh. "Lifetime Use of Nursing Home Care". The New England Journal of Medicine, Vol. 324, No.9 (1991), [9] Kessler, enis. "The Long-Term Care Insurance Market". The Geneva Paers on Risk and Insurance- Issues and Practice, Vol. 33 (2008), [10] Norton, Edward C. "Long-Term Care", in A.J. Culyer and J.P. Newhouse (Eds): Handbook of Health Economics, Vol. 1, Chater 17 (2000),

35 EU B B=0 0 EU γ γ=0 < 0 π(1 π) EU B B=0 0 B=0 v I B=0 π B B=0 B=0 + δ B=0 π(1 π)δ( L Q) EU γ γ=0 < 0 γ=0 v I γ=0 < π γ γ=0 γ=0 + δ γ=0

36 W W B=0 = W γ=0 v I γ=0 v γ=0 I B=0 = v I π(1 π) δ( L Q)π B B=0 B=0 + δ B=0 γ=0 γ=0 v I γ=0 B=0 v I B=0 = π B B=0 B=0 + δ < π γ γ=0 γ=0 + δ γ=0 B=0 B=0 = γ=0 δ( L Q) W B=0 = B B=0 < γ=0 γ γ=0 δ( L Q) < γ γ=0 B B=0 γ γ=0 = δ Q γ=0 + γ=0 δ(1 π)( L Q) r I r B B=0 = B=0 (1 π) I f I r γ=0 = γ γ=0 B B=0 = I f B=0 Q δ r γ=0 I f B=0 (1 π) + δ( L Q) EU γ γ=0 0 EU B B=0 > 0 EU γ γ=0 0 π(1 π)δ( L Q) γ=0 v I γ=0 π γ γ=0 γ=0 + δ γ=0

37 π(1 π) π(1 π) EU B B=0 > 0 B=0 v I B=0 > π B B=0 γ=0 v W I π γ=0 δ( L Q) B=0 + δ W B=0 γ γ=0 γ=0 + δ γ=0 π δ( L Q) γ γ=0 γ=0 + δ γ=0 > π B B=0 B=0 + δ B=0 1 δ( L Q) γ γ=0 > B B=0 γ γ=0 B B=0 > δ( L Q)

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