Social long-term care insurance with two-sided altruism

Transcription

1 IDEI July 2015 Social long-term care insurance wit two-sided altruism Helmut Cremer, Pierre Pestieau, Kerstin Roeder 7

2 Social long-term care insurance wit two-sided altruism 1 Helmut Cremer 2, Pierre Pestieau 3 and Kerstin Roeder 4 July Te autors gratefully acknowledge nancial support from te Caire "Marcé des risques et création de valeur" of te FDR/SCOR, 2 Toulouse Scool of Economics (GREMAQ, IDEI and Institut universitaire de France), 21 Allée de Brienne, Toulouse, France; elmut.cremer@tse-fr.eu. 3 CREPP, Université de Liège; CORE, Université catolique de Louvain, and TSE. 4 University of Augsburg, Germany.

3 Abstract Tis paper studies te design of a social long-term care (LTC) insurance wen altruism is two-sided. Te laissez-faire solution is not e cient, unless tere is perfect altruism. Under full information, te rst-best can be decentralized by a linear subsidy on informal aid, a linear tax on bequests wen te parent is dependent and state speci c lump-sum transfers wic provide insurance. We also study a second-best sceme comprising a LTC bene t, a payroll tax on cildren s earnings and an ineritance tax. Tis sceme redistributes resources across individuals and between te states of nature and te tax on cildren s labor enances informal care to compensate for te cildren s possible less tan full altruism. Keywords: long-term care, social insurance, two-sided altruism JEL: H2, H5

4 1 Introduction Long-term care (LTC) is becoming a major concern for policy makers. Following te rapid aging of our societies, te needs for LTC are expected to grow and yet tere is a lot of uncertainty as ow to nance tose needs; see Norton (2009) and Cremer, Pestieau and Pontière (2012) for an overview. Family solidarity, wic as been te main provider of LTC, is reacing a ceiling, and te market is remains rater tin. 1 Not surprisingly, one would expect tat te state takes te relay. Tis paper studies te design of social LTC insurance wen bot parents and cildren are altruistic towards eac oter. Cildren s altruism is owever only partial. First, it is only triggered by te occurrence of dependency and second it is limited. In a pure market economy te dependent parents ave to devote teir resources to purcase professional care services, but also to leave some bequests to boost informal care. One cannot be but struck by two parallel evolutions: te soaring needs for LTC and te growing sare of inerited wealt in overall capital accumulation; see Piketty and Zucman (2014). LTC is surely not a problem for te very wealty ouseolds, but for many ouseolds LTC can eat up most of teir assets and make it impossible to bequeat anyting. It is tus not surprising tat some people ave tougt of using te proceeds of estate taxation to nance public long-term care. An example of tis is te Englis green paper proposing a voluntary ineritance levy. Accordingly, people in England and Wales could pay a one-o ineritance levy of up to 12,000 in return for free public long-term care in teir old age. Te fee would eiter be deducted from te estates of older people wen tey die or be paid wen tey enter retirement age. 2 1 Te economic literature as identi ed a number of factors tat can explain te low level of LTC insurance demand. Tese include signi cant loading factors due to ig administrative costs, adverse selection in te demand for insurance (Sloan and Norton, 1997; Finkelstein and McGarry, 2006; Brown and Finkelstein, 2007 and 2009) and te existence of ceaper substitutes like family care or public assistance (Pauly, 1990; Zweifel and Strüwe, 1998; Brown and Finkelstein, 2008). 2 See ttp:// accessed 20t July

5 Te aim is to avoid forcing many pensioners to sell teir family omes to fund massive nursing ome bills. Tere is indeed a close link between bequests and long-term care. In a world witout a well-developed LTC insurance market, ouseolds are forced to oversave or to selfinsure. Consequently, in case of good ealt tey end up wit an excess of saving tat can lead to involuntary bequests. It is ten clear tat if it were possible to tax tose bequests, te proceeds could be used to nance long-term care. Actually if some type of insurance, private or public were available, one would end up wit te same result, tat is, a perfect smooting of consumption between te two states of te world, dependency or not. Brunner and Pec (2012) made tat argument by sowing tat a tax on bequests to nance LTC causes a smaller deadweigt loss tan an income or consumption tax. Tis olds true weter or not te parents are altruistic. Te problem is tat in a world wit eterogeneous agents it migt be impossible to distinguis bequests tat are made by dependent parents from tose made by autonomous parents. Te matter gets more complicated wen we look at te case of formal LTC nanced by saving or insurance and informal services coming from cildren. For te time being we assume tat formal and informal care are complements. Dependent parents will ave to excange tose informal services against te prospect of some bequests. If cildren are not altruistic, we will ave a pure quid pro quo excange; if tey are altruistic tey migt elp teir parents even if tese cannot bequeat anyting. Te amount of elp will depend on te extent of lial altruism. In case of perfect altruism, cildren will provide te optimal amount of assistance to teir parents. In a world of identical individuals, te social insurance sceme will serve two purposes: it redistributes resources across te two states of nature and it induces te cild to elp is parent. If individuals di er in te level of teir wage, social insurance must also redistribute resources across individuals. To study tese issues we consider a population consisting of one parent and one 2

6 cild families. Parents are pure altruists towards teir cild, wile te cild s altruism may be imperfect. Parents are retired and ave accumulated some wealt. Tey face a probability of becoming dependent and needing LTC. Te need of LTC requires expenditures of some monetary amount. In case of dependency parents would like to bene t from te aid of teir cildren, wo are ready to elp teir parent out of altruism but also wit te expectation of some ineritance. 3 We caracterize te rst-best allocation and sow tat te laissez-faire is not ef- cient, unless tere is perfect altruism. Under full information te rst-best can be decentralized by a linear subsidy on informal aid, a linear tax on bequests wen te parent is dependent, and state speci c lump-sum transfers wic provide insurance. Next, we study te second-best allocation wen te instruments available are linear (state independent) taxes on bequests and cildren s labor income wic nance a transfer to te dependent elderly. 4 Observe tat te tax on cildren s labor income is in our setting e ectively equivalent to a subsidy on informal care. 5 We rst consider a setting wit ex ante identical individuals. We sow tat bot taxes sould be positive. Te tax on labor wic subsidizes informal care compensates for possible imperfect altruism; tis is like in te rst-best implementation. Bot taxes also provide insurance, and tat is relevant as te rst-best state speci c lump-sum transfers or taxes tat provide full insurance are not available. Labor and bequest taxes are ten used to provide (partial) insurance. Finally, we consider a setting wit ex ante eterogenous families, wic di er in cildren s productivities and parents wealt. We sow tat te results obtained in te omogenous family case carry over. However, te two tax rates now also include a positive redistributive term. Trougout tis paper private LTC insurance is assumed 3 See Pauly (1990) for te preference parents ave for assistance from teir cildren. 4 For non-linear scemes, see Jousten et al. (2006) and Pestieau and Sato (2009). 5 Labor and informal care are te sole possible usages of cildren s time. Tere is no leisure or te amount of leisure is given. 3

7 away. Tis is for te sake of simplicity but also because in most countries te LTC insurance market is extremely tin; see Brown and Finkelstein (2007, 2009). 2 Identical individuals 2.1 Te model setup Consider a population of a size normalized to one, consisting of one parent (subscript p ) and one cild (subscript c ) families. Parents are pure altruists towards teir cild, wile a cild s altruism may be imperfect. Parents are retired and ave accumulated wealt y. Tey face te probability of becoming dependent and needing long-term care. Te need of LTC requires expenditures of amount L. In case of dependency te parent would like to bene t from te aid of is cild, wo is ready to elp is parent out of altruism but also wen receiving some ineritance b. Parents coose te bequest to teir cildren. In case of autonomy, te cild inelastically supplies one unit of labor at a wage rate w. In case te parent needs long-term care, te cild s time spend on te labor market is reduced by te time spend for informal care provision a and gross earnings amount to w(1 a). Care provided by te cild reduces te monetary loss from LTC by (a) L (wit 0 > 0; 00 < 0) since ten te parent requires less professional care services. In sum, te parent s and cild s expected utilities are given by EU p = [H(y + (a) L b) + u(w(1 a) + b)] i + (1 ) u(y b b) + u(w + b b) ; (1) EU c = [u(w(1 a) + b) + H(y + (a) L b)] + (1 ) u(w + b i b) + u(y b b) ; (2) were 2 [0; 1] re ects te cild s degree of altruism and a b indicates te state of staying ealty. Te utility function u is te standard one wereas, H denotes te utility 4

8 for long-term care. Tese functions satisfy u 0 ; H 0 > 0 and u 00 ; H 00 < 0. Furtermore, we assume H(x) < u(x) for any x, due to te loss of autonomy. Te timing of te model is as follows: te government moves rst and announces its policy (stage 0). Ten, te parent and te cild play te following two-stage game. In stage 1, te parent cooses (for eac state of nature) a level of bequest. In stage 2, after te state of nature is revealed, cildren decide ow muc informal care to provide if teir parent is dependent. Oterwise, cildren simply consume teir income plus te bequest. To determine te subgame perfect Nas equilibrium, we solve tis game by backward induction. We rst study te rst-best allocation wic provides a bencmark against wic we can compare te laissez-faire allocation, obtained by dropping stage First-best solution Wit ex ante identical families, we can de ne te optimal allocation as te one maximizing te expected utility of a representative dynasty. Assuming tat bot parents and cildren receive equal social weigts, te rst-best problem can be written as max m;c; bm;bc;a W = [H(m) + u(c)] + (1 ) [u( bm) + u(bc)] ; s.t. (1 ) [ bm + bc L] + [m + c] = y + (1 )w + [w(1 a) + (a)] ; (3) were te decision variables are informal care provision a and parent s and cild s consumption in bot states of nature. We denote te latter by m, bm, c and bc respectively. In te rst-best all variables are directly set, assuming full information and disregarding te multi-stage structure of te game. Te speci cation of te game will of course be relevant below, wen we study te decentralization of te rst-best optimum. Let denote te Lagrange multiplier associated wit te resource constraint (3). Te rst 5

9 order conditions (FOCs) caracterizing te optimal solution can be written as follows H 0 (m) = u 0 ( bm) = u 0 (c) = u 0 (bc) = ; (4) w = 0 (a): (5) Equation (4) states te equality of marginal utilities of incomes across generations and states of nature (full insurance), wile equation (5) describes te e cient coice of informal care. It states tat te opportunity cost of informal care w equals its marginal bene t. 2.3 Laissez-faire allocation Stage 2: Coice of cildren In case of autonomy, cildren do not ave to make any decision. Tey consume teir income and bequest, enjoying a utility u(w + b b). Wen teir parents are in need of LTC, tey solve te following problem max a u(w(1 a) + b) + H(y L + (a) b): Te rst-order condition of te above problem is given by wu 0 (w(1 a) + b) + H 0 (y L + (a) b) 0 (a) = 0: Tis equation de nes te optimal level of a as a function of b: a = a(b). We ave as te second-order condition (SOC) = wu00 (c) + H 00 (m) 0 (a) SOC a > 0; (6) SOC a = w 2 u 00 (c) + H 0 (m) 00 (a) + H 00 (m)( 0 (a)) 2 < 0: In words, if parents increase teir bequest, te willingness to provide informal care by te cild also increases. 6

10 2.3.2 Stage 1: Coice of parents Te problem of te parents is to coose (in bot states of nature) te level of bequests to teir cildren. In case of dependency parents ave more needs but at te same time tey migt leave a iger bequest to induce more assistance from teir cildren. Teir problem is te following max b; b b EU p = [H(y L + (a ) b) + u(w(1 a ) + b)] i + (1 ) u(y b b) + u(w + b b) : (7) Taking into consideration equation (6) te FOCs wit respect to b and b b can be written as H 0 (m) 1 0 (a (1 ) + u 0 (c) = (8) u 0 ( bm) + u 0 (bc) = 0: (9) It can be easily veri ed tat for = 1 te parent leaves te e cient amount of bequests so tat H 0 (m) = u 0 (c). Tis in turn goes and in and wit an e cient amount of aid provided by te cild 0 (a) = w; see equation (5). Tis result is not surprising as for perfect altruism, on bot te parent s and te cild s side, te optimization problem of te two family members coincides wit te one of te social planner. Wenever < 1 tis is no longer te case as te cild puts a too low value on te parent s bene ts of informal care. In tis case te parent leaves a large tan oterwise e cient bequest to induce te cild to provide more care. 2.4 Decentralization of te rst-best allocation Assume for te time being tat tere is no asymmetry of information so tat all relevant variables including informal aid are publicly observable. In te following we sow tat te FB allocation witin our multi-stage setting can be decentralized by a lump-sum 7

11 transfer (social LTC insurance) from te ealty to te dependent elderly ( b D; D) supplemented by a tax on labor income a and on bequests b for tose cildren wose parents are dependent. budget constraint Public transfers must be cosen suc tat te government s D = (1 ) b D + [ a w(1 a ) + b b ] (10) is balanced Family s problem reconsidered Again, in case of autonomy cildren do not ave to make any decision; tey consume teir income and bequest enjoying a utility u(w + b b). Wen teir parents are in need of long-term care, tey now solve te following problem max a u((1 a )w(1 a) + (1 b )b) + H(y + D L + (a) b): (11) Te rst order condition of te above problem is given by (1 a )wu 0 (c) + H 0 (m) 0 (a) = 0: (12) Te above equation yields a = a(b; a ; b ; D). Te comparative static e ects wrt. b and te policy instruments =(1 a)wu 00 (c)(1 b ) + H 0 (m) 0 (a) SOC a > 0; = wu0 (c) (1 a )w 2 (1 a)u 00 a SOC a 7 0; = (1 b)wbu 00 b SOC a < = H0 (m) 0 (a) SOC a < 0: (16) Care increases wit bequests wile a tax on bequest dampens it. An income tax eiter puses or dampens informal care provision and public LTC decreases informal care. 6 Te second order condition is SOC a = ((1 a)w) 2 u 00 (c) + H 0 (m) 00 (a) + H 00 (m)( 02 < 0. 8

12 Comparison of equation (12) wit te rst-best allocation (equations (4) and (5)) sows tat to implement te rst-best level of care we rst need a tax on cild s income, i.e. a = 1. A tax on income implicitly subsidizes te cild s informal care provision so tat te trade-o between te cild s marginal costs and marginal bene t of care is te e cient one. Te problem for te parents is now te following max b; b b U p = [H(y + D L + (a ) b) + u((1 a )w(1 a ) + (1 b )b)] i + (1 ) u(y D b b b) + u(w + b b) : (17) Using equation (12), te FOCs wit respect to b, and b are H 0 (m) 1 0 (a (1 ) + b )u 0 (c) = 0; (18) u 0 ( bm) + u 0 (bc) = 0: (19) Equations (18) and (19) yield b = b(d; b D; a ; b ) and b b = b b(d; b D; a ; b ). Wen te transfers (D; b D) are cosen suc tat te parent (and tus also te cild) is fully insured implying H 0 (m) = u 0 ( bm) we ave H 0 (m) = u 0 (bc). Te tax on bequests must ten be cosen suc tat its e ect on informal care is o set, i.e. b = 0 (a (1 ) > Tat is, we ave to tax bequests wen te parent ends up being dependent. Tis tax o sets te parent s incentives to induce informal care above its rst-best level. 7 To decentralize te rst-best optimum we need tree types of tools: a distortionary tax a tat fosters cild s assistance in case of impure altruism ( < 1); a tax b on bequests wen te parent is dependent and lump-sum transfers acting as redistributive device between te two states of nature (D; b D). 7 Te expression =@b is not independent of b. 9

13 Note tat we ave assumed away private insurance. If private insurance were available, its role would depend on te existence and te size of loading costs. Wit zero loading costs, a rst best could be acieved simply by taxing cild s labor at te appropriate rate, namely 1, and by taxing bequests wen te parent is dependent. Wit positive loading cost, we also need te lump-sum transfers, wic ten completely crowd out private insurance. 3 Second-best solution 3.1 Identical cildren We now turn to te second-best. We assume tat te government can use only linear (proportional) taxes on bequests and on labor earnings t, to nance a LTC bene t g. Tax rates cannot be conditioned on te parents ealt status. We use te notation t, and g rater ten a, b and D to avoid confusion wit te rst-best implementation. Wit our instruments we write te revenue constraint as g = t( Te cild s problem Te cild solves a )w + b + (1 ) b b i : max a u ((1 t)w(1 a) + (1 )b) + H (y L b + g (a)) ; (20) wic apart from te cange in notation is equivalent, to (11) so tat te FOC continues to be given by (12) and can be written as (1 t)wu 0 (c) = H 0 (m) 0 (a ): (21) Te solution is given by a = a(b; t; ; g). Te comparative statics properties of tis function follow tose described by equations (13) (16). Using subscripts to denote partial derivatives, we tus ave a b > 0; a < 0; a t? 0; a g < 0. 10

14 3.1.2 Parent s problem Turning to te parent s problem, it is given by max b; b b EU p = [H (y L b + g + (a )) + u ((1 t)w(1 a ) + (1 )b)] + (1 ) u(y b b) + u (1 t)w + (1 ) b i : (22) Te two FOCs wrt. b b and b are u 0 ( bm) = u 0 (bc) (1 ); (23) H 0 (m) 1 0 (a )a b = u 0 (c)[(1 ) (1 t)wa b ]: (24) Using (21), equation (24) can be rewritten as H 0 (m) 1 0 (a )(1 )a b = u 0 (c)(1 ): Tese conditions de ne te bequests supply functions, i.e. b = b(t; ; g) and b = b(t; ). We implicitly assume positive bequests in eiter state of nature. Substituting for b into a yields te level of aid as a function of te government s instruments a = a (b(t; ; g); t; ; g) ~a(t; ; g): Te government s problem Te second-best problem can now be expressed by te following Lagrangean L(g; t; ) = [H (y L b + g + (a )) + u ((1 t)w(1 a ) + (1 )b )] + (1 ) u y b + u (1 t)w + (1 ) b i g t(1 a )w (b + (1 ) b i ) (25) 11

15 Using te envelope teorem, we can write te FOCs wrt. g, t = H 0 (m) 0 (a ) u 0 (c)(1 t)w ~a g + H 0 (m) [ + tw~a g b g ] = H 0 (m) 0 (a ) u 0 (c)(1 t)w ~a t u 0 (c) (1 a) + (1 )u 0 (bc) w + w(1 a ) tw~a t + b t + (1 ) b ii t = H 0 (m) 0 (a ) u 0 (c)(1 t)w ~a u 0 (c)b + (1 )u 0 (bc) b i + b + (1 ) b tw~a + b + (1 ) b ii = 0: Tese FOCs can be rearranged using te compensated form wit dg d = b + (1 ) b ; and dg dt = (1 a )w : In oter words, we keep te tax rates t and as sole decision variables, wile accounting of course for teir impact on g, via te government s budget constraint. Tis yields c @ d = (1 )H0 (m) 0 (a )~a c u 0 (c)b + (1 )u 0 (bc) b i + H 0 (m)(b + (1 ) b ) tw~a c b c + (1 ) b i c = 0; dt = (1 )H0 (m) 0 (a )~a c t u 0 (c) (1 a ) + (1 )u 0 (bc) w + H 0 (m)(1 a )w tw~a c t b c t + (1 ) b i c t = 0: (27) To interpret tese formulas, we make te standard assumption tat te cross-derivatives are negligible. Formally, b c t = b b c t = ~a c = 0. Ten, we ave 8 We de ne ~a c x = b (1 ) bb b c + (1 ) b i ; (28) c t = (1 )H0 (m) 0 (a )~a c t y (1 ) w b w~a c ; (29) t dg dx bc @g dg and b c dx x dg, were x = ; t. dx 12

16 were u 0 (c) H 0 (m)? 0 and b u 0 (bc) H 0 (m) < 0. 9 Te denominators in tese two formulas are standard. Tey re ect te e ciency e ect of taxes. Normally one expects ~a c t > 0 and b c + (1 ) b c < 0: To interpret te numerator of equation (28), we ave to return to our assumption of positive bequests and on te inequalities u 0 (c) H 0 (m)? 0 and b u 0 (bc) H 0 (m) < 0. As sown above in te rst-best, = b = 0. Te numerator will tus be positive if is not too ig and if b > > 0; wic is likely if L is large enoug. Turning to te numerator of equation (29), note tat te rst term in te nominator vanises if = 1. Oterwise, it represents te need to foster cild s assistance. Tis term mirrors te rst-best implementation but as a di erent structure in te secondbest. Te second and tird terms are te same as in te previous condition. Tey re ect te insurance bene ts provided by te tax. Tey give te welfare gap between te young and te dependent elderly in te two states of nature. If te marginal utility of te dependent is ig relative to tat of te young, ten tere is a need for a iger tax to nance te social bene t g. Private insurance would not cange our qualitative results. Even wit zero loading cost, te rst-best cannot be acieved wit te instruments considered ere; it would require a state-contingent ineritance tax. Wit a positive loading cost individuals buy less tan full insurance and te two taxes continue to make up for part of te incompleteness of private insurance protection. 3.2 Heterogenous families We now assume tat families di er ex ante in te cildren s productivity levels and in te parents resources. We assume tat bot (w i and y i ) are positively correlated so tat our assumption of positive bequests is valid across families. Again, te government 9 Tis means tat bm > bc > c. 13

17 levies a at tax t on earnings and on bequests to nance a uniform LTC bene t g. Witin suc a framework te social insurance sceme pursues an additional objective, namely redistribution. Te government s revenue constraint as to be modi ed suc tat g = X i n i t(1 a i )w i + (b i + (1 ) b i i ) ; were n i is te relative number of families wit productivity w i and y i. Te optimizing beavior of cildren and parents remains te same as described above; eac cild and parent of type-i solves te problem given in (20) and (22). From tis we obtain optimal long-term care provision a i = ~a(t; ; g) wit ~a i < 0; ~a it? 0 and ~a ig < 0 and optimal bequests b i = b(t; ; g) and b b i = b(t; ): 3.3 Government s problem Te second-best problem of a utilitarian government wit eterogenous families can be expressed by te following Lagrangean expression L(g; t; ) = X i n i nh (y i L b i + g + (a i )) + u ((1 t)w i (1 a i ) + (1 )b i ) + (1 ) y i b i + u u g t(1 a i )w i (1 t)w i + (1 ) b i b i + (1 ) b i i i o : (30) 14

18 Maximizing tis expression wit respect to g, t and yields te following =E [ H 0 (m) 0 (a ) u 0 (c)(1 t)w ~a g + H 0 (m) o [ + tw~a g b g ] = =E H 0 (m) 0 (a ) u 0 (c)(1 t)w ~a t u 0 (c) (1 a ) + (1 )u 0 (bc) w + w(1 a ) tw~a a + b t + (1 ) b i o t = =E H 0 (m) 0 (a ) u 0 (c)(1 t)w ~a u 0 (c)b + (1 )u 0 (bc) b i + b + (1 ) b tw~a + b + (1 ) b i o = 0; were we use te operator E for P i n i. As for identical families we can rearrange te above FOCs in terms of compensations wit dg=d and dg=dt; see Appendix (A.2). By assuming tat te cross derivatives are nil, i.e. b c it = b b c it = ~ac i = 0, we ave = cov(; b ) (1 ) cov( ; b b ) E b c + (1 ) b i c + EEb (1 )EE b b E b c + (1 ) b i ; (31) c and t = (1 )E [H0 (m) 0 (a )~a c t] cov(; w(1 a )) (1 ) cov( b ; w) Ew~a c t + EEw(1 a ) (1 )E b Ew Ew~a c t (32) Te interpretation of tese two formulas is te same as in te case of identical individuals wit one exception, namely te covariance terms tat are expected to be negative. Tis is because wit iger income/wealt te di erence in marginal utilities between states of nature decreases. In oter words, earnings inequality puses for more taxation of eiter earnings or bequests. 15

19 4 Conclusion Te starting point of tis paper is te concomitance of two trends: te increasing needs for LTC, and unusual growt of ineritance. In a laissez faire economy wit neiter public nor private insurance, dependency would bite a big cunk of accumulated wealt wit te consequence tat bequests would greatly di er for kids wit dependent parents and kids wit ealty parents. Cildren s care can only partially mitigate tose di erences. Given tat te market for LTC insurance is typically tin or even absent, one as to rely on public action to restore some balance between te two states of nature. In a rst-best world, a subsidy on cildren s care, state-dependent bequest taxes and a state-contingent lump sum tax could acieve te optimum and provide full insurance. If for obvious reasons individualized lump sum taxes are not available, one as to rely on second-best scemes. In tis paper a wage tax and a linear ineritance tax contribute to nance a uniform LTC bene t. Te main conclusion is tat under plausible conditions linear taxes on bot ineritances and wages to nance a public LTC at rate bene t are desirable. Te tax on cildren s labor income e ectively subsidizes informal care, like in te rst-best implementation. It is required wen altruism is less tan full. Bot taxes also provide insurance and, in te case of eterogenous families (di ering in cildren s productivity and parent s wealt) redistributive bene ts. We ave seen tat we can end up wit a subsidy of bequests or even earnings in te special case were te welfare of te dependent would be quite ig relative to tat of is cildren and te probability of dependency would be particularly ig. Trougout tis paper we ave assumed tat parents in bot states of nature were leaving some bequests. Tis implies tat parents are relatively wealtier tan teir cildren, and tat dependency does not represent a large nancial loss. In an extension 16

20 of tis work we would like to explore cases were some parents do not leave any bequest or tey only bequeat in te ealty state. It is expected tat if bequests in te ealty state were muc iger tan tose in te state of dependency and if te wealty families were leaving ig bequests wereas te poorer families were bequest-constrained, estate taxation would play an even more important role. Among oter extensions, we would like to introduce private insurance wit loading costs and study te possibility of nonlinear LTC policies. A Appendix A.1 Identical families: second-best Using a sorter notation, we can write u 0 (c)b + (1 )u 0 (bc) b b i + H 0 (m)(b + (1 ) b b ) = b (1 ) b b b u 0 (c) (1 a ) + (1 )u 0 (bc) w + H 0 (m)(1 a )w = (1 a )w (1 ) b w We now rewrite equations (26) and (27) as (1 )H 0 (m) 0 (a )~a c b (1 ) bb = tw~a c (1 )H 0 (m) 0 (a )~a c t (1 a )w (1 ) b w = A.2 Heterogenous families: second-best b c + (1 tw~a c t ) b c b c t + (1 i ; ) b b c t i c =E n (1 )H 0 (m) 0 (a )~a c + H 0 (m)(b + (1 ) b b ) u 0 (c)b + (1 )u 0 (bc) b b i tw~a c b c + (1 ) b iio c = 0; (33) =E (1 )H 0 (m) 0 (a )~a c t u 0 (c) (1 a ) + (1 )u 0 (bc) w + H 0 (m)(1 a )w tw~a c t b c t + (1 ) b iio c t = 0: (34) We use u 0 (c) H 0 (m)? 0 and b u 0 (bc) H 0 (m) < 0; to represent te gap between te cild s marginal utility of consumption and tat of te dependent parent. 17

21 We now rewrite te FOC s as (1 )E H 0 (m) 0 (a )~a c t cov(; w(1 a )) (1 ) cov( ; b w) EEw(1 a ) (1 )EEw b = E tw~a c t b c t + (1 ) b c t (1 )E H 0 (m) 0 (a )~a c cov(; b ) (1 ) cov( ; b b ) EEb (1 )E b E b b = E tw~a c b c + (1 ) b b c ii ; (35) ii : (36) Solving for t and yields te expressions in te text. References [1] Brown, J.R. and A. Finkelstein, (2007), Wy is te Market for Long-Term Care Insurance so Small?, Journal of Public Economics, 91, [2] Brown, J.R. and A. Finkelstein, (2008), Te Interaction of Public and Private Insurance: Medicaid and te Long-Term Care Insurance Market, American Economic Review, 98, [3] Brown, J.R. and A. Finkelstein, (2009), Te Private Market for Long-Term Care Insurance in te United States: a Review of te Evidence, Journal of Risk and Insurance, 76, [4] Brunner, J. and S. Pec, (2012), Te Bequest Tax as Long-Term Care Insurance, Te Scandinavian Journal of Economics, 114, [5] Cremer, H., Pestieau, P., and G. Pontière, (2012), Te Economics of Long- Term-Care: A Survey, Nordic Economic Policy Review, 2, [6] Finkelstein, A. and K. McGarry, (2006), Multiple Dimensions of Private Information: Evidence from te Long-Term Care Insurance Market, Te American Economic Review, 96,

22 [7] Jousten, A., Lipszyc, B., Marcand, M., and P. Pestieau, (2005), Long- Term Care Insurance and Optimal Taxation for Altruistic Cildren, FinanzArciv - Public Finance Analysis, 61, [8] Norton, E., (2000), Long-Term Care, in Cuyler, A., and J. Newouse (Eds.), Handbook of Healt Economics, Volume 1b, Capter 17. [9] Pauly, M.V., (1990), Te Rational Nonpurcase of Long-Term Care Insurance, Journal of Political Economy, 1990, 98 (1), [10] Pestieau, P., and M. Sato, (2009), Social and Private Long-Term Care Insurance wit Variable Altruism, unpublised manuscript. [11] Piketty, T., and G. Zucman, (2014), Wealt and Ineritance in te Long-Run, in Handbook of Income Distribution, Nort Holland, Vol. 2. [12] Sloan, F.A. and E.C. Norton, (1997), Adverse Selection, Bequests, Crowding Out, and Private Demand for Insurance: Evidence from te Long-Term Care Insurance Market, Journal of Risk and Uncertainty, 15, [13] Zweifel, P. and W. Strüwe, (1998), Long-Term Care Insurance in a Two- Generation Model, Journal of Risk and Uncertainty, 65,