Introduction Long-term

∗

∗

Introduction

Long-term care has been increasingly becoming a "hot" topic over the last years, and its importance is predicted to grow even more in the coming decades.

Long-term care is the care for people who are dependent on the help 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 provided both formally (by paid professional caregivers) and informally (by family members or friends), at home and in special institutions.

The need for long-term care is highly related with age. For instance, around half of all long-term care users in the OECD countries are over 80 years old (OECD, 2011). Therefore, more and more attention to long-term care issues is required due to the apparent ageing of the populations in most industrialised countries. Indeed, it is expected that in the OECD the share of the population aged 80 and over will increase from 4% in 2010 to 9.4% in 2050 (com pared to 1%

in 1950) (OECD, 2011). The European Union has estimated potential increases in the number of dependent old persons under different scenarios and predicts that, even assuming a decrease in age-specific disability rates in the future, the number of dependent old people in the EU27 will rise by 90% from 2007 to 2060 (European Commission, 2009). If the disability rates by age remain constant, the predicted increase is even more drastic and reaches 115%, which means that the number of dependent elderly is expected to more than double (European Commission, 2009).

In the light of these trends, it is important 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 provided and financed by three institutions: the family, the State and the market (Cremer et al., 2012). Most care is provided by family caregivers. Even though it is difficult to obtain precise 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 providers (for example, in the Netherlands, Canada, United States, New Zealand) (OECD, 2011). However, it is not clear whether the current situation will last. The soci- etal changes such as the rising number of childless families, the increasing female participation in the labour market, the mobility of children, higher divorce rates or changing family values are likely to decrease the availability of family help (Cremer et al., 2012; OECD, 2011). Therefore, with the drastically rising de- mand for long-term care, it is becoming increasingly important to ensure reliable alternatives to family solidarity.

State intervention can be seen as one of the solutions. In most industrialised countries, the governments participate in the provision and/or financing of long- term care services. However, the extent and the nature of this participation is rather different across countries. In addition to this, very few countries have explicit social long-term care insurance schemes (Cremer et al., 2012; OECD,

2

2011). In the presence of the current demographic and societal trends, there is thus a need to assess the suitability of the current policy designs and to look for the best ways of public intervention, which among other things should take into account its possible effects on the family and the market.

Turning to the market, another source of help in the problem of dependence could be private long-term care insurance. However, here we are facing another issue, namely, the so-called long-term care insurance puzzle. This puzzle has been created by a surprisingly low demand for private long-term care insurance.

In particular, even though the potential long-term care costs are large

¹

and the probability to become dependent is high,

²

only a small fraction of individuals purchase private long-term care insurance. A number of different factors poten- tially explaining this puzzle have been proposed in the literature. Those possible explanations range from the ones that assume perfect rationality of individuals (for instance, the high price of insurance or crowding out by the State) to the ones considering the non-purchase of insurance as irrational (such as myopia or ignorance and denial of the issue of dependence).

³

Even though it seems that none of the proposed reasons alone is able to entirely explain the puzzle, each of them might be playing some role in the issue. Therefore, being able to solve or at least reduce (some of) these potential problems might help to stimulate the market for private long-term care insurance.

This paper is concerned with one of these potential problems. In particular, it is the problem of intra-family moral hazard, which was proposed as a possible explanation for the long-term care insurance puzzle by Pauly (1990). The idea of the intra-family moral hazard argument is the following. In the case of de- pendence, expenditures on formal long-term care reduce the dependent person's wealth (and thus the bequest that will be left to his/her children). The children can help to avoid these formal care expenses (and thus protect the future be- quest) by providing care to their parent themselves. However, if the parent has long-term care insurance, the cost of formal care is (at least partly) covered by the insurer. Thus, long-term care insurance also protects the parent's wealth (and the future bequest),

⁴

which results in the children having less incentive to provide care themselves. Therefore, a parent who prefers 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 developed with little formalization in Pauly (1990), Zweifel and Striiwe (1998) provide 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-purchase

1

For instance, a nursing home stay in the U.S. costs between $40 000 and $70 000 per year, while the average cost in France is around € 35 000 per year (Taleyson, 2003).

2

According to Kemper and Murtaugh (1991), a 65-year-old person has a probability of 43% to enter a nursing home at some time before his/her death (this probability is 52% for women and 33% for men). See also Norton (2000).

3

For recent surveys of potential explanations for the puzzle, see Cremer et al. (2012), Pestieau and Ponthiere (2011) and Brown and Finkelstein (2011).

4

Pauly (1990, 1996) names the protection of one's bequest as the main function of long-term care insurance.

3

of long-term care insurance.

⁵

However, the analysis of Zweifel and Struwe (1998) only focuses on one type of insurance benefits, namely, benefits which are proportional to the amount of long-term care expenditures. The present paper puts forward the idea that the extent of the above described problem might be different depending on the form of insurance benefits.

The proportional benefits analyzed by Zweifel and Struwe (1998) roughly capture the idea behind long-term care insurance benefits prevalent in the U.S where these benefits take the form of a reimbursement for the long-term care expenditures the insured has, in the limit of a certain reimbursement ceiling (Duran and Taleyson, 2003). However, this is not the only type of long-term care insurance benefits used in practice. For instance, private insurers in France use a substantially different benefit scheme. In particular, insurance payments in France are of the form of a fixed monthly cash benefit which starts being paid if the insured is recognized to be dependent. This benefit does not depend on the amount of long-term care expenditures the person has and can be used by the person in the way he/she decides. This clearly contrasts with the U.S. where, in order to be reimbursed, the long-term care expenditures have to come from the use of services that are approved by the insurer (Duran 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 help to limit intra-family moral hazard. Some empirical 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 consider- ations (such as intentions to ensure sufficiently large bequests to their children) that parents could have in their insurance purchase decisions. Nevertheless, in- terestingly enough, this does not seem to be the case in France. In particular, a study conducted by Courbage and Roudaut (2008) finds empirical evidence that in France long-term care insurance purchases are driven by altruistic motives.

First, they find that the demand for long-term care insurance rises with the probability of leaving a bequest, which confirms Pauly's (1990, 1996) argument that long-term care insurance is purchased to protect one's bequest. Further, they find that long-term care insurance is positively associated with the number of children and the fact of living with a partner. Finally, it is found that long-

5

A recent paper by Courbage and Zweifel (2011) suggests to look at intra-family moral hazard as at a two-sided phenomenon arguing that apart from the just described effect of insurance on the incentives for children (one side of the phenomenon), the second side exists in that parents may purchase less insurance if they can rely on the caregiving effort of their children. However, the caregiving effort in their paper is modeled as a preventive one that helps to keep the parent out of a nursing home (which more generally can be seen as actually preventing from a need for long-term care). In the present paper, I consider the children's caregiving as the provision of care when the parents' need for long-term care has already materialized and focus on the first side of the phenomenon (the incentives for children), as suggested by Pauly (1990) and Zweifel and Striiwe (1998).

4

term care insurance is positively associated with the probability of receiving informal care in the case of such a need in the future. These findings suggest that people are not afraid that insurance coverage might distort caregiving in- centives 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 comparison, 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 relationship between stated altruism (i.e. the person's wish to leave a bequest) and demand for long- term care insurance. This suggests that a parent might well be willing to leave a bequest, but still decide not to purchase long-term care insurance. While it is possible that some other reasons exist, one explanation for this could be the attempt to avoid intra-family moral hazard.

Therefore, one could suspect that intra-family moral hazard is a more serious problem in the U.S. than in France. While it is possible that some other factors come into play as well, a potential explanation for this difference could be the different nature of insurance benefits.

The aim of this paper is to explore more formally the roles of fixed and proportional insurance benefits in the issue of intra-family moral hazard. In particular, in the context of intra-family moral hazard, the paper compares the impact of fixed and of proportional insurance benefits on the long-term care related behavior of children and eventually on the insurance purchase decisions of parents.

The model considers an elderly parent and his adult child. The parent might become dependent, in which case he places a special value on the long-term care received from his child. The parent also cares about the bequest that will be left to the child; thus, he might want to purchase insurance in order to protect his bequest from long-term care expenses. If the parent becomes dependent, the child chooses the amount of care she wants to provide. I consider separately the cases when the child likes and dislikes providing care to the parent. In addition to the amount of caregiving, the child's utility depends on her wealth which consists of her earnings in the labour market and the bequest left by the parent.

The bequest is the channel through which the child's care provision is affected by the insurance coverage purchased by the parent before the materialization of the risk of dependence. Reasoning backwards, the first part of the analysis studies the problem of the child and compares the effects that insurance with fixed and proportional benefits has on the child's choice of caregiving. The second part of the analysis explores how these different effects (anticipated by the parent) influence the parent's insurance purchase 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 propor- tional benefits and thus do not make any comparisons. Also, the model in this paper differs from theirs in a number of other aspects such as, for instance, the way the proportional benefits are modeled. In addition, Zweifel and Striiwe (1998) model the parent and the child as the principal and the agent. I do not adopt this approach here but rather follow the idea of equal treatment of both

5

parties suggested by Courbage and Zweifel (2011).

⁶

It should be also mentioned that, differently from Zweifel and Striiwe (1998), the present paper looks at the problem of the child from the perspective of the standard consumer's theory, which allows to identify and understand the substitution and income effects that influence the child's choice.

The paper finds that the amount of caregiving of a child who dislikes pro- viding care is decreasing in the parent's insurance coverage both in the case of proportional and in the case of fixed insurance benefits. Thus, intra-family moral hazard exists with both types of benefits. However, it is also shown that the same amount of long-term care insurance purchased by the parent results in more care given by the child when insurance benefits are fixed than when they are proportional. This confirms that fixed insurance benefits indeed help to mitigate the phenomenon of intra-family moral hazard. On the other hand, intra-family moral hazard is never a problem with fixed insurance benefits when the child likes providing care. In that case, fixed benefits not only eliminate intra-family moral hazard but also trigger an opposite effect, namely, an in- crease in the child's caregiving. With proportional benefits, both an increase and a decrease in the child's care provision is possible, which means that intra- family moral hazard cannot be completely ruled out. In addition, even in the case when intra-family moral hazard is not a problem with either type of ben- efits, it is shown that the child's chosen amount of caregiving is greater with fixed than with proportional insurance benefits. It is also interesting to note that the child's utility is higher with fixed rather than with proportional ben- efits. Turning to the parent, 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 parent will find insurance beneficial, purchasing a positive amount of insurance is more likely to be in the parent's interest when benefits are fixed rather than proportional.

The results of the paper indeed confirm that a low degree of intra-family moral hazard in France could (at least partly) be attributed to the use of fixed insurance benefits. Moreover, the results could also partly explain the relative success of the French private long-term care insurance market. While the French and the American markets are the world's two most developed markets for private long-term care insurance, in 2010, about 15% of the population aged 40 and over had private long-term care insurance in France, compared to only 5%

in the U.S. (OECD, 2011). Given the findings of the paper, it seems reasonable to believe that the use of fixed benefits takes part in the success of France.

Indeed, if fixed benefits (through the softening of intra-family moral hazard) can make it more likely that a parent will decide to purchase 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 parent's choice of insurance and the child's choice of care provision are modeled as simultaneous decisions. Here the choices are sequential since the parent has to make the insurance decision first (before one of the two possible 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

p c π 1 − π

EU ^p = πU _D ^p + (1 − π)U _I ^p

U _D ^p = u ^p (Q, W _p ^D ) U _I ^p =

v ^p (X _p ^I , W _p ^I )

X _p ^I W _p ^I

L ¯

L, ¯ −∞

L, ¯ + ∞ L ¯

(Q Q

∂u ^p /∂Q > 0

W _p ^D

δ γ ∈ [0, 1]

L ¯ − Q

πγδ( ¯ L − Q) W ¯ p

W _p ^D = ¯ W p − πγδ( ¯ L − Q) − (1 − γ)δ( ¯ L − Q)

W _p ^I = ¯ W p − πγδ( ¯ L − Q) − X _p ^I

L ¯

Q

γ γ

B πB

W _p ^D = ¯ W p − πB − δ( ¯ L − Q) + B

W _p ^I = ¯ W p − πB − X _p ^I

B B

U _D ^c = u ^c (Q, W _c ^D )

Q (W _c ^D )

∂u^c

∂Q

< 0

∂u^c

∂Q

> 0

W _c ^D = w( ¯ L − Q) + W _p ^D

w ( ¯ L − Q)

L ¯

πγδ( ¯ L − Q) Q.

Q

Q

γ Q γ

Q(γ) πγδ( ¯ L − Q(γ))

Q(γ)

∂u^c

∂Q

< 0

∂u^c

∂Q

> 0

^∂u_∂Q^c

< 0

∂u

^c

∂Q < 0

γ B

w

w ≥ δ

(1 − γ)δ ≤ w < δ

Q = 0

Q

w

w < (1 − γ)δ

w

Q,W max

_c^D

U _D ^c = u ^c (Q, W _c ^D )

s.t. W _c ^D = w( ¯ L − Q) + ¯ W p − πγδ( ¯ L − Q) − (1 − γ)δ( ¯ L − Q)

⇐⇒

W _c ^D = ¯ W p − πγδ( ¯ L − Q) − [(1 − γ)δ − w]( ¯ L − Q)

⇐⇒

W _c ^D − [(1 − γ)δ − w]Q = ¯ W p − πγδ( ¯ L − Q) − [(1 − γ)δ − w] ¯ L

Q W _c ^D (1 − γ)δ > w [(1 − γ)δ − w]

p pr

W _c ^D − p pr Q = ¯ W p − πγδ( ¯ L − Q) − p pr L ¯

� �� �

I

pr

πγδ( ¯ L − Q)

I pr

Q,W max

_c^D

U _D ^c = u ^c (Q, W _c ^D )

s.t. W _c ^D − p f Q = ¯ W p − πB + B − p f L ¯

� �� �

I

f

p f = δ − w > p pr

Q p pr I pr p f I f

∂Q

∂p i

= ∂ Q ˜

∂p i

+ Q ∂Q

∂I i

i = pr, f Q ˜

∂

Q ˜

∂pi

p i

∂u^c

∂Q

< 0 L ¯ − Q

∂Q

∂Ii

< 0 Q.

∂Q(p pr , I pr )

∂γ = ∂Q

∂p pr

∂p pr

∂γ + ∂Q

∂I pr

∂I pr

∂γ = − δ

� ∂ Q ˜

∂p pr

+ Q ∂Q

∂I pr

� +

+ ∂Q

∂I pr

�

− πδ( ¯ L − Q) + πγδ ∂Q

∂γ + δ L ¯

�

⇐⇒

∂Q

∂γ =

(

−

)

� �� �

− δ ∂ Q ˜

∂p pr

+

� (+) �� � Q ∂Q

∂I pr

( − δ) +

� (−) �� �

∂Q

∂I pr

[δ L(1 ¯ − π) + πδQ]

1 − ∂Q

∂I pr

πγδ

� �� �

(+)

∂Q

∂γ

∂Q

∂γ

I pr

∂Q

∂γ =

(

−

)

� �� �

− δ ∂ Q ˜

∂p pr

+

� (−) �� �

∂Q

∂I pr

δ(1 − π)( ¯ L − Q) 1 − ∂Q

∂I pr

πγδ

� �� �

(+)

< 0

∂Q(p f , I f )

∂B = ∂Q

∂p f

∂p f

���� ∂B

=0

+ ∂Q

∂I f

∂I f

∂B = ∂Q

∂I f

����

(−)

(1 − π) < 0

w < (1 − γ)δ

B

d[γδ( ¯ L − Q)] = 1

�

δ( ¯ L − Q) − γδ ∂Q

∂γ

� dγ = 1

⇐⇒

dγ = 1

δ( ¯ L − Q) − γδ

^∂Q_∂γ

∂Q

∂B = ∂Q

∂I f

(1 − π)

∂Q

∂γ · dγ | d[γδ( ¯ L

−

Q)]=1 = − δ

_∂p^∂

^{Q ˜}

_pr

+

_∂I^∂Q_pr

δ(1 − π)( ¯ L − Q)

1 −

∂I^∂Qpr

πγδ · 1

δ( ¯ L − Q) − γδ

^∂Q_∂γ

∂Q

∂γ

· dγ | d[γδ( ¯ L

−

Q)]=1

∂Q

∂B

� ∂Q

∂γ · dγ | d[γδ( ¯ L

−

Q)]=1

�

|

^γ=0

= − 1

( ¯ L − Q) · ∂ Q ˜

∂p pr |

^γ=0

+ ∂Q

∂I pr |

^γ=0

(1 − π)

∂Q

∂B | ^B=0 = ∂Q

∂I f | ^B=0 (1 − π)

∂Q

∂Ipr

|

^γ=0

=

_∂I^∂Q

f

| ^B=0

w < (1 − γ)δ

w < (1 − γ)δ

L ¯ − Q)

∂u^c

∂Q

< 0)

− [(1 − γ)δ − w]

( ¯ L − Q) ^pr πγδ( ¯ L − Q) ^pr

πγδ( ¯ L − Q) ^pr

− [δ − w]

γδ( ¯ L − Q) ^pr

B B =

γδ( ¯ L − Q) ^pr

− [δ − w]

( ¯ L − Q) ^f < ( ¯ L − Q) ^pr

∂u

^c

∂Q > 0

w

w ≤ (1 − γ)δ Q = ¯ L)

(1 − γ)δ < w ≤ δ Q = ¯ L

w > δ

w

Q,W max

_c^D

U _D ^c = u ^c (Q, W _c ^D )

s.t. W _c ^D = w( ¯ L − Q) + ¯ W p − πγδ( ¯ L − Q) − (1 − γ)δ( ¯ L − Q)

⇐⇒

W _c ^D = ¯ W p − πγδ( ¯ L − Q) + [w − (1 − γ)δ]( ¯ L − Q)

⇐⇒

W _c ^D + [w − (1 − γ)δ]Q = ¯ W p − πγδ( ¯ L − Q) + [w − (1 − γ)δ] ¯ L

w > (1 − γ)δ [w − (1 − γ)δ]

ρ pr

W _c ^D + ρ pr Q = ¯ W p − πγδ( ¯ L − Q) + ρ pr L ¯

� �� �

M

pr

M pr

Q,W max

_c^D

U _D ^c = u ^c (Q, W _c ^D )

s.t. W _c ^D + ρ f Q = ¯ W p − πB + B + ρ f L ¯

� �� �

M

f

ρ f = w − δ < ρ pr

Q ρ pr M pr

ρ f M f

∂Q

∂ρ i

= ∂ Q ˜

∂ρ i − Q ∂Q

∂M i

i = pr, f Q ˜

∂

Q ˜

∂ρi

ρ i

∂Q

∂Mi

> 0

Q.

∂Q(ρ pr , M pr )

∂γ = ∂Q

∂ρ pr

∂ρ pr

∂γ + ∂Q

∂M pr

∂M pr

∂γ = δ

� ∂ Q ˜

∂ρ pr − Q ∂Q

∂M pr

� +

+ ∂Q

∂M pr

�

− πδ( ¯ L − Q) + πγδ ∂Q

∂γ + δ L ¯

�

⇐⇒

∂Q

∂γ =

(

−

)

� �� � δ ∂ Q ˜

∂ρ pr

� (−) �� �

− Q ∂Q

∂M pr

δ +

� (+) �� �

∂Q

∂M pr

[δ L(1 ¯ − π) + πδQ]

1 − ∂Q

∂M pr

πγδ

� �� �

(?)

∂Q

∂γ

M pr

∂Q

∂γ =

(

−

)

� �� � δ ∂ Q ˜

∂ρ pr

+

� (+) �� �

∂Q

∂M pr

δ(1 − π)( ¯ L − Q) 1 − ∂Q

∂M pr

πγδ

� �� �

(?)

� 0

∂Q(ρ f , M f )

∂B = ∂Q

∂ρ f

∂ρ f

���� ∂B

=0

+ ∂Q

∂M f

∂M f

∂B = ∂Q

∂M f

� �� �

(+)

(1 − π) > 0

w > δ

� ∂Q

∂γ · dγ | d[γδ( ¯ L

−

Q)]=1

�

|

^γ=0

= 1

( ¯ L − Q) · ∂ Q ˜

∂ρ pr |

^γ=0

� �� �

(−)

+ ∂Q

∂M pr |

^γ=0

� �� �

(+)

(1 − π)

∂Q

∂B | ^B=0 = ∂Q

∂M f | ^B=0

� �� �

(+)

(1 − π)

− _{( ¯ L−Q)} ¹ · ∂ Q ˜

∂ρ pr |

^γ=0

� �� �

(−)

> ∂Q

∂M pr |

^γ=0

� �� �

(+)

(1 − π),

− _{( ¯ L}

₋

¹ _Q) · ∂ Q ˜

∂ρ pr |

^γ=0

� �� �

(

−

)

= ∂Q

∂M pr |

^γ=0

� �� �

(+)

(1 − π)

− _{( ¯ L−Q)} ¹ · ∂ Q ˜

∂ρ pr |

^γ=0

� �� �

(

−

)

< ∂Q

∂M pr |

^γ=0

� �� �

(+)

(1 − π),

∂Q

∂Mpr

|

^γ=0

=

_∂M^∂Q_f

| ^B=0

w > δ

w > δ

− [w − (1 − γ)δ]

Q ^pr

πγδ( ¯ L − Q ^pr )

πγδ( ¯ L − Q ^pr )

− [w − δ]

γδ( ¯ L − Q ^pr )

B B =

γδ( ¯ L − Q ^pr )

− [w − δ]

Q ^f > Q ^pr

only eliminate this phenomenon but also trigger an opposite effect, namely, an increase in the child's caregiving. It is true that an increase in the child's care provision is possible with proportional 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 problem, fixed insurance benefits still allow the parent to enjoy more care given by his child.

Summing up the results of this section, it first should be noted that the impact of insurance on the child's caregiving is quite different in the two cases of the child's preferences analyzed. In the case when the child dislikes providing care (and has a low wage), insurance unambiguously decreases the amount of her caregiving with both types of insurance benefits. In the case when the child likes providing care (and has a high wage), with fixed insurance benefits, the effect is always opposite: insurance increases the child's caregiving. With proportional benefits, the effect of insurance cannot be unambiguously determined. However, irrespective of whether we look at Case 1 or Case 2, it is clear that, when the amount of insurance is the same with both types of benefits, the child's care provision is greater when insurance benefits are fixed. It is also interesting to note that, again independently of the case analyzed, the child achieves more utility with fixed rather than with proportional insurance benefits.

We now turn to the analysis of the parent.

3. Fixed vs Proportional: The impact on the par- ent's insurance decision

Having compared the impact of insurance on the child's care provision in the cases of fixed and proportional benefits, we can now see how this affects the parent's insurance purchase decisions. In this section, I first compare the parent's expected utility with fixed and with proportional insurance benefits and then analyze in more detail the parent's decision of whether to buy long-term care insurance or not. As in the previous section, I discuss separately the cases when the child likes and dislikes providing care.

3.1. Parent's expected utility

This subsection looks at the parent's expected utility with the two types of insurance benefits. Let us start with the case when the child dislikes providing care.

23

∂u

^c

∂Q < 0

EU ^p = πU _D ^p + (1 − π)U _I ^p

U _D ^p = u ^p (Q, W _p ^D ) U _I ^p =

v ^p (X _p ^I , W _p ^I )

γ w < (1 − γ)δ

γ w < (1 − γ)δ Q ^pr

γδ( ¯ L − Q ^pr ) B γδ( ¯ L − Q ^pr )

W _p ^D Q

Q ^f > Q ^pr Q ^f B = γδ( ¯ L − Q ^pr )

W _p ^D = ¯ W p − πγδ( ¯ L − Q ^pr ) − (1 − γ)δ( ¯ L − Q ^pr ) =

= ¯ W p − πγδ( ¯ L − Q ^pr )

� �� �

=πB

− δ( ¯ L − Q ^pr ) + γδ( ¯ L − Q ^pr )

� �� �

=B

W _p ^D = ¯ W p − πB − δ( ¯ L − Q ^f ) + B Q ^f > Q ^pr

Q W _p ^D

W _p ^I = ¯ W p − πγδ( ¯ L − Q ^pr )

� �� �

=πB

− X _p ^I

W _p ^I = ¯ W p − πB − X _p ^I

(1 − γ)δ ≤ w < δ

w < δ

w ≥ δ

∂u

^c

∂Q > 0

w > δ

w > δ

∂u

^c

∂Q < 0

max

γ, X_p^I

EU ^p = πu ^p (Q, W _p ^D ) + (1 − π)v ^p (X _p ^I , W _p ^I )

γ

X _p ^I X _p ^I

∂v ^p

∂X _p ^I + ∂v ^p

∂W _p ^I

∂W _p ^I

∂X _p ^I = 0

⇐⇒

∂v ^p

∂X _p ^I − ∂v ^p

∂W _p ^I = 0 γ

∂EU ^p

∂γ = π

� ∂u ^p

∂Q

∂Q

∂γ + ∂u ^p

∂W _p ^D

∂W _p ^D

∂γ

�

+ (1 − π)

� ∂v ^p

∂X _p ^I

∂X _p ^I

∂γ + ∂v ^p

∂W _p ^I

∂W _p ^I

∂γ

�

=

= π

� ∂u ^p

∂Q

∂Q

∂γ +

�

δ( ¯ L − Q)(1 − π) − γδ ∂Q

∂γ (1 − π) + δ ∂Q

∂γ

� ∂u ^p

∂W _p ^D

� +

+(1 − π)

� πγδ ∂Q

∂γ − πδ( ¯ L − Q)

� ∂v ^p

∂W _p ^I

B, X max

_p^I

EU ^p = πu ^p (Q, W _p ^D ) + (1 − π)v ^p (X _p ^I , W _p ^I )

B X _p ^I

B

∂EU ^p

∂B = π

� ∂u ^p

∂Q

∂Q

∂B + ∂u ^p

∂W _p ^D

∂W _p ^D

∂B

�

+ (1 − π)

� ∂v ^p

∂X _p ^I

∂X _p ^I

∂B + ∂v ^p

∂W _p ^I

∂W _p ^I

∂B

�

=

= π

� ∂u ^p

∂Q

∂Q

∂B +

�

(1 − π) + δ ∂Q

∂B

� ∂u ^p

∂W _p ^D

� +

+(1 − π) [ − π] ∂v ^p

∂W _p ^I

∂EU^p

∂γ

∂EU^p

∂B

∂EU ^p

∂γ |

^γ=0

= π(1 − π)δ( ¯ L − Q)

� �� �

(+)

� ∂u ^p

∂W _p ^D |

^γ=0

− ∂v ^p

∂W _p ^I |

^γ=0

�

� �� �

(+)

+π ∂Q

∂γ |

^γ=0

� �� �

(

−

)

� ∂u ^p

∂Q |

^γ=0

+ δ ∂u ^p

∂W _p ^D |

^γ=0

�

� �� �

(+)

∂EU ^p

∂B | ^B=0 = π(1 − π)

� �� �

(+)

� ∂u ^p

∂W _p ^D | ^B=0 − ∂v ^p

∂W _p ^I | ^B=0

�

� �� �

(+)

+π ∂Q

∂B | ^B=0

� �� �

(

−

)

� ∂u ^p

∂Q | ^B=0 + δ ∂u ^p

∂W _p ^D | ^B=0

�

� �� �

(+)

γ γ = 0

w < δ w < (1 − γ)δ

Q Q

Q

∂u^p

∂W_p^D

|

^γ=0

�

=

_∂W^∂upD p

| ^B=0 �

−

∂W^∂vp_p^I

|

^γ=0

�

=

_∂W^∂vpI p

| ^B=0 �

> 0 Q

Q Q

v

^p

(X

^I_p

, W

_p^I

) = lnX

_p^I

+ lnW

_p^{I ∂u}

p(Q, W_p^D)

∂W_p^D

=

_W¹D

p

u

^c

(Q, W

_c^D

) =

lnW

_c^D

− Q W ¯

p

>

^2δ(δ_δ+w^−w)

w < (1 − γ)δ

� ∂EU ^p

∂γ dγ | d[γδ( ¯ L

−

Q)]=1

�

|

^γ=0

= π(1 − π)

� �� �

(+)

� ∂u ^p

∂W _p ^D |

^γ=0

− ∂v ^p

∂W _p ^I |

^γ=0

�

� �� �

(+)

+

+π

� ∂Q

∂γ · dγ | d[γδ( ¯ L−Q)]=1

�

|

^γ=0

� �� �

(−)

� ∂u ^p

∂Q |

^γ=0

+ δ ∂u ^p

∂W _p ^D |

^γ=0

�

� �� �

(+)

∂EU ^p

∂B | ^B=0 = π(1 − π)

� �� �

(+)

� ∂u ^p

∂W _p ^D | ^B=0 − ∂v ^p

∂W _p ^I | ^B=0

�

� �� �

(+)

+π ∂Q

∂B | ^B=0

� �� �

(−)

� ∂u ^p

∂Q | ^B=0 + δ ∂u ^p

∂W _p ^D | ^B=0

�

� �� �

(+)

∂u^p

∂W_p^D

|

^γ=0

−

∂W^∂vp_p^I

|

^γ=0

=

_∂W^∂upD

p

| ^B=0 −

∂W^∂vp_p^I

| ^B=0

∂u^p

∂Q

|

^γ=0

+ δ

_∂W^∂upD

p

|

^γ=0

=

^∂u_∂Q^p

| ^B=0 + δ

_∂W^∂upD p

| ^B=0

γ

w < (1 − γ)δ

∂EU^p

∂B

| ^B=0 ≤ 0

^∂EU_∂γ^p

|

^γ=0

< 0

∂EU^p

∂γ

|

^γ=0

≥ 0

^∂EU_∂B^p

| ^B=0 > 0

∂u

^c

∂Q > 0

w > δ

∂EU^p

∂γ

|

^{γ=0 ∂EU}∂B^p

| ^B=0

∂EU ^p

∂γ |

^γ=0

= π(1 − π)δ( ¯ L − Q)

� �� �

(+)

� ∂u ^p

∂W _p ^D |

^γ=0

− ∂v ^p

∂W _p ^I |

^γ=0

�

� �� �

(+)

+π ∂Q

∂γ |

^γ=0

� �� �

(+)/(

−

)

� ∂u ^p

∂Q |

^γ=0

+ δ ∂u ^p

∂W _p ^D |

^γ=0

�

� �� �

(+)

∂EU ^p

∂B | ^B=0 = π(1 − π)

� �� �

(+)

� ∂u ^p

∂W _p ^D | ^B=0 − ∂v ^p

∂W _p ^I | ^B=0

�

� �� �

(+)

+π ∂Q

∂B | ^B=0

� �� �

(+)

� ∂u ^p

∂Q | ^B=0 + δ ∂u ^p

∂W _p ^D | ^B=0

�

� �� �

(+)

∂EU^p

∂B

| ^B=0

Q

∂Q

∂γ

|

^γ=0

≥ 0

∂Q

∂γ

|

^γ=0

< 0

∂EU^p

∂γ

|

^γ=0

w > δ

might have a somewhat different impact on the behavior of children and parents than benefits proportional to long-term care expenditures do. In particular, the paper has studied and compared the effects that insurance with fixed and proportional benefits has on the caregiving choice of the child and, consequently, the influence that these different effects have on the insurance purchase decisions of the parent who values the care provided by his child.

The results of the paper formally confirm the intuitive hypothesis that fixed insurance benefits should help to limit the phenomenon of intra-family moral hazard. Indeed, in the case of a child who dislikes providing care, the paper finds that, even though the child's caregiving is decreasing in the insurance coverage with both types of benefits, the same amount of long-term care insurance pur- chased by the parent results in more care provided by the child when insurance benefits are fixed than when they are proportional. Thus, with fixed benefits the child reduces her caregiving less drastically. The amount of the child's care provision is greater with fixed than with proportional benefits in the case when she likes providing care as well. In that case, fixed benefits even eliminate intra- family moral hazard completely, whereas the effect of proportional benefits is ambiguous.

The fact that the child provides more care when benefits are fixed rather than proportional makes the parent prefer insurance with fixed benefits: the same amount of insurance gives him a higher expected utility when benefits are fixed. However, the parent will not always be willing to buy a positive amount of insurance even when fixed benefits are available; nevertheless, it is shown that purchasing a positive amount of insurance is more likely to be in the parent's interest when benefits are fixed rather than proportional. In other words, the non-purchase of long-term care insurance is more likely with proportional benefit schemes, which could partly explain why private long-term care insurance is purchased by a larger share of the targeted population in France than in the U.S.

Besides adding to the explanation of the success of the French insurance scheme, the findings of the paper can be of some use to public policy as well.

Zweifel and Striiwe (1998) have raised concerns about the welfare effects that are/could be caused by a compulsory social long-term care insurance. In partic- ular, since (because of intra-family moral hazard) parents might in many cases find private long-term care insurance undesirable, imposing a compulsory so- cial insurance might go against the interests of many parents. The analysis in this paper has shown that insurance can be made more desirable (or less un- desirable) to parents by using fixed benefit schemes. Indeed, it is shown that insurance is more likely to be in the interest of the parent when benefits are fixed rather than proportional. Also, even in the cases when fixed benefits do not help to make insurance desirable to the parent, his expected utility is still higher with fixed than with proportional benefits. In addition to this, the paper also finds that the child is better-off with fixed benefits as well. Therefore, based on these results, it seems reasonable to favour public schemes with fixed rather than proportional benefits.

33

Finally, it should be reminded that, as mentioned in the introduction, the analysis in this paper is focused on the context of intra-family moral hazard and thus abstracts from other factors and situations that could potentially modify the conclusions about the desirability of fixed and proportional insurance ben- efits. Thus, rather than drawing general inferences, the paper provides conclu- sions based on a particular context which, nevertheless, constitutes a potential piece of the long-term care insurance puzzle and is therefore important to un- derstand.

References

[1] Brown, Jeffrey R. and Amy Finkelstein. "Insuring Long-Term Care in the United States". Journal of Economic Perspectives, Vol. 25, No. 4 (2011), pp. 119-142.

[2] Courbage, Christophe and Nolwenn Roudaut. "Empirical Evidence on Long-Term Care Insurance Purchase in France". The Geneva Papers on Risk and Insurance- Issues and Practice, Vol. 33 (2008), pp. 645-658.

[3] Courbage, Christophe and Peter Zweifel. "Two-Sided Intergenerational Moral Hazard, Long-Term Care Insurance, and Nursing Home Use". Jour- nal of Risk and Uncertainty, Vol. 43 (2011), pp. 65-80.

[4] Cremer, Helmuth, Pierre Pestieau and Gregory Ponthiere. "The Economics of Long-Term Care: A Survey". CORE Discussion Paper (2012).

[5] Cutler, David. "Why Doesn't the Market Fully Insure Long-Term Care?''.

NBER Working Paper No. 4301 (1993).

[6] Duran, Romain and Lucie Taleyson. "Les Raisons du Succes de l' Assurance Dependance en France". Risques - Les Cahiers de l' Assurance, Vol. 55 (2003), pp. 115-120.

[7] European Commission. "The 2009 Ageing Report: Economic and Bud- getary Projections for the EU-27 Member States (2008-2060)". Joint Report prepared by the European Commission (DG ECFIN) and the Economic Policy Committee (AWG), 2009.

[8] Kemper, Peter and Christopher M. Murtaugh. "Lifetime Use of Nursing Home Care". The New England Journal of Medicine, Vol. 324, No.9 (1991), pp. 595-600.

[9] Kessler, Denis. "The Long-Term Care Insurance Market". The Geneva Pa- pers on Risk and Insurance- Issues and Practice, Vol. 33 (2008), pp. 33-40.

[10] Norton, Edward C. "Long-Term Care", in A.J. Culyer and J.P. Newhouse (Eds): Handbook of Health Economics, Vol. 1, Chapter 17 (2000), pp.

955-994.

34

∂EU^p

∂B

| ^B=0 ≤ 0

^∂EU_∂γ^p

|

^γ=0

< 0

∂EU^p

∂B

| ^B=0 ≤ 0 π(1 − π)

� ∂u ^p

∂W _p ^D | ^B=0 − ∂v ^p

∂W _p ^I | ^B=0

�

≤ − π ∂Q

∂B | ^B=0

� ∂u ^p

∂Q | ^B=0 + δ ∂u ^p

∂W _p ^D | ^B=0

�

∂EU^p

∂γ

|

^γ=0

< 0 π(1 − π)δ( ¯ L − Q)

� ∂u ^p

∂W _p ^D |

^γ=0

− ∂v ^p

∂W _p ^I |

^γ=0

�

< − π ∂Q

∂γ |

^γ=0

� ∂u ^p

∂Q |

^γ=0

+ δ ∂u ^p

∂W _p ^D |

^γ=0

�

∂u^p

∂W_p^D

| ^B=0 =

_∂W^∂upD

p

|

^γ=0 ∂W^∂vp_p^I

| ^B=0 =

_∂W^∂vpI

p

|

^γ=0

�

∂u^p

∂W_p^D

| ^B=0 −

∂W^∂vp_p^I

| ^B=0 �

� =

∂u^p

∂W_p^D

|

^γ=0

−

∂W^∂vp_p^I

|

^γ=0

�

π(1 − π) �

∂u^p

∂W_p^D

|

^γ=0

−

∂W^∂vp_p^I

|

^γ=0

�

− π

^∂Q_∂B

| ^B=0 �

∂u^p

∂Q

| ^B=0 + δ

_∂W^∂upD p

| ^B=0 �

− δ( ¯ L − Q)π ∂Q

∂B | ^B=0

� ∂u ^p

∂Q | ^B=0 + δ ∂u ^p

∂W _p ^D | ^B=0

�

< − π ∂Q

∂γ |

^γ=0

� ∂u ^p

∂Q |

^γ=0

+ δ ∂u ^p

∂W _p ^D |

^γ=0

�

∂u^p

∂Q

| ^B=0 =

^∂u_∂Q^p

|

^γ=0 ∂W^∂up_p^D

| ^B=0 =

_∂W^∂upD p

|

^γ=0

δ( ¯ L − Q)

�

− ∂Q

∂B | ^B=0

�

<

�

− ∂Q

∂γ |

^γ=0

�

⇐⇒

δ( ¯ L − Q) < ∂Q

∂γ |

^γ=0

� ∂Q

∂B | ^B=0

∂Q

∂γ |

^γ=0

= − δ ∂ Q ˜

∂p pr |

^γ=0

+ ∂Q

∂I pr |

^γ=0

δ(1 − π)( ¯ L − Q)

∂Q

∂B | ^B=0 = ∂Q

∂I f | ^B=0 (1 − π)

∂Q

∂Ipr

|

^γ=0

=

_∂I^∂Q_f

| ^B=0

∂Q

∂γ |

^γ=0

� ∂Q

∂B | ^B=0 =



 − δ

_∂p^∂

^{Q ˜}

_pr

|

^γ=0

∂Q

∂If

| ^B=0 (1 − π)





� �� �

(+)

+ δ( ¯ L − Q)

∂EU^p

∂γ

|

^γ=0

≥ 0

^∂EU_∂B^p

| ^B=0 > 0

∂EU^p

∂γ

|

^γ=0

≥ 0 π(1 − π)δ( ¯ L − Q)

� ∂u ^p

∂W _p ^D |

^γ=0

− ∂v ^p

∂W _p ^I |

^γ=0

�

≥ − π ∂Q

∂γ |

^γ=0

� ∂u ^p

∂Q |

^γ=0

+ δ ∂u ^p

∂W _p ^D |

^γ=0

�

∂EU^p

∂B

| ^B=0 > 0 π(1 − π)

� ∂u ^p

∂W _p ^D | ^B=0 − ∂v ^p

∂W _p ^I | ^B=0

�

> − π ∂Q

∂B | ^B=0

� ∂u ^p

∂Q | ^B=0 + δ ∂u ^p

∂W _p ^D | ^B=0

�

π(1 − π)

� ∂u ^p

∂W _p ^D |

^γ=0

− ∂v ^p

∂W _p ^I |

^γ=0

�

≥ − π δ( ¯ L − Q)

∂Q

∂γ |

^γ=0

� ∂u ^p

∂Q |

^γ=0

+ δ ∂u ^p

∂W _p ^D |

^γ=0

�

− π δ( ¯ L − Q)

∂Q

∂γ |

^γ=0

� ∂u ^p

∂Q |

^γ=0

+ δ ∂u ^p

∂W _p ^D |

^γ=0

�

> − π ∂Q

∂B | ^B=0

� ∂u ^p

∂Q | ^B=0 + δ ∂u ^p

∂W _p ^D | ^B=0

�

− 1 δ( ¯ L − Q)

∂Q

∂γ |

^γ=0

> − ∂Q

∂B | ^B=0

⇐⇒

∂Q

∂γ |

^γ=0

� ∂Q

∂B | ^B=0 > δ( ¯ L − Q)

�