# Long-Term Care Insurance and Family Norms

• Chiara Canta and Pierre Pestieau

Abstract: Long-term care (LTC) is mainly provided by the family and subsidiarily by the market and the government. To understand the role of these three institutions, it is important to understand the motives and the working of family solidarity. In this paper, we focus on the case when LTC is provided by children to their dependent parents out of some norm that has been inculcated to them during their childhood by some exemplary behavior of their parents towards their own parents. In the first part, we look at the interaction between the family and the market in providing for LTC. The key parameters are the probability of dependence, the probability of having a norm-abiding child and the loading factor. In the second part, we introduce the government which has a double mission: correct for a prevailing externality and redistribute resources across heterogeneous households.

## Appendix A: Proof of Proposition 1

We will first prove the existence and uniqueness of a stationary equilibrium, in which for all t. Then we will show that this is indeed the unique equilibrium of the intergenerational game.

### Stationary equilibrium: existence

The stationary equilibrium is given by the fixed point of the best response function implicitly defined in [5]. Since and are continuous functions, is also continuous. Furthermore, is convex and compact. Then , has a fixed point by Brouwer’s theorem, and there exists a stationary equilibrium of the intergenerational game.

Setting in [5] yields

Since is strictly concave, this expression implicitly defines the unique fixed point of . Thus, the intergenerational game admits a unique stationary equilibrium, with defined by [6].

Under our assumption that is strictly greater than zero. Furthermore, is strictly smaller than one. To see this, it is sufficient to verify that the first order condition [2] at is strictly negative. This is always the case if satisfies the standard Inada condition . Consequently, .

### Uniqueness

The first order condition with respect to can be rewritten as

Using the implicit function theorem, we can write

Thus, the best response function is monotonically increasing. Furthermore, it is easy to show that and , so that setting equal to zero or one is never a best response for any generation.

Since has a unique fixed point, it has to cross the identity line from above. Thus, if and only if , where is the fixed point of . Conversely, if and only if .

Given these features of the best response function, we can show that there does not exist any equilibrium such that at least one generation chooses a family norm different from . We will consider two cases.

1. Suppose that there exists an equilibrium such that . Since is an equilibrium strategy, it is a best response to . Due to the monotonicity of the best response function, implies . Repeating this argument, it is possible to prove that the best responses of generations satisfy . For an n high enough, . Thus, in equilibrium . However, setting the family norm equal to one is never a best response, so that this cannot be an equilibrium strategy of the intergenerational game.

2. A similar reasoning can be applied for the case . If this is an equilibrium strategy, then for an n high enough, . However, setting the family norm equal to zero is never a best response, so that this cannot be an equilibrium of the intergenerational game.

## Appendix B: Proof of Proposition 2

The first-order conditions with respect to and B are

and

Using the envelope theorem and observing that either , or (implying ), we can rewrite the conditions above as

[12]

and

[13]

Substituting [12] in [13], we get

so that consumption is smoothed across states. Furthermore,

Since the individual first-order condition with respect to is , we can rewrite this condition as

## Appendix C: comparative statics with respect to w

Equation [6] permits us to recover . Total derivation of [3] and [4] yields

In order to solve this system, define

and

Using this notation, we can write

and

Straightforward calculations yield (under the assumption that )

and

Therefore, savings increase in the productivity parameter, while the sign of is ambiguous.

## Appendix D: Proof of Proposition 6

The Lagrange expression for the planning problem is

where is the Lagrangian multiplier associated with the revenue constraint. The first-order conditions with respect to and B yield

[14]

and

[15]

Using the above first-order conditions, we can write

[16]

where is obtained from the resource constraint of the government. We define

The sign of is ambiguous whenever public insurance crowds out private. Combining [14] and [15], we can rewrite [16] as

After simplifications, this expression yields [11].

## Acknowledgements

We are very grateful to Helmut Cremer, Justina Klimaviciute and Dirk Van de gaer for useful comments and suggestions. We also wish to thank participants to seminars in University of Liège, KUL and to the Journées LAGV 2012 and the 2012 CESifo Venice Summer Institute Workshop on the Economics of Long-Term Care. We aknowledge financial support from the chair “Marché des risques et création de valeur” of the FdR/SCOR, as well as the Belgian Program on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Minister’s Office, Science Policy Programming. The scientific responsibility is assumed by the authors.

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1. 1

The present paper will not consider LTC for younger individuals, rather it will focus on LTC of the elderly.

2. 2

Eurostat, EUROPOP2008 convergence scenario of the 27 member states. See also the European Union 2009 Ageing Report.

3. 3
4. 4
5. 5

There exist a number of papers studying exchanges within the family. See, e.g., Stern and Engers (2002). For surveys, see Norton (2000) and Grabowsky, Norton Houtven and Van (2012).

6. 6

Pezzin, Pollak, Schone (2009) study couples where one of the spouses is disabled. The nondisabled spouses is more likely to provide care to the disabled one if the couple has children. This evidence is in line with a demonstration effect at work.

7. 7

We adopt an “asexual” setting in which each young adult makes individual choices. We thus abstract from the decision process within the household. In reality, it is clear that the choice of is made at the level of the household.

8. 8

An interesting extension would be to endogenize ρ, for example, by making it dependent on the behavior of the parents.

9. 9

An alternative specification could have been that the individual provides aid of length just in case of dependency of his parent, with the expectation that in case of his own dependency, he would get . This specification happened to be more complex analytically. Furthermore, such a modeling strategy would not be compatible with the demonstration effect: only children whose grandparents were dependent would be exposed to a family norm.

10. 10

All along, we assume interior solutions, i.e.,

11. 11

This result is in line with Cox and Stark (1996), who find that the number of contacts with elderly parents decreases in the children’s income.

12. 12

Alternatively, one could analyze the case in which the social planner is able to impose a mutualization of family help. In such a case, is never wasted, since individuals with healthy parents are forced to help the dependent elderly not belonging to their family. This specification would be more relevant for traditional societies with extended families. Our model applies to nuclear families.

13. 13

If instead children made a monetary investment in family help, this would not affect their productivity. In this case, a lump-sum transfer, instead than a payroll tax, would be necessary to implement the optimal level of family norm.

14. 14

This is not necessarily true, since public LTC insurance also discourages savings, and savings and family norm are substitutes.

15. 15

If , individuals purchase full LTC insurance on the private market. In this case, , irrespective of the level of B. Thus, public LTC insurance cannot affect the level of the family norm, and the optimal B is equal to zero.

16. 16

If , individuals purchase full insurance LTC on the private market. Thus, the first term in [8] is equal to zero. A tax decentralizes the first-best family norm. Then, a public LTC benefit financed by a payroll tax decentralizes the first best whenever private LTC insurance is actuarially fair.

Published Online: 2013-4-30

©2014 by Walter de Gruyter Berlin / Boston