Family altruism and long-term care

Family altruism and long-term care

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Abstract
This article studies the Nash equilibrium of a simultaneous game with two players: a
dependent elderly person (the parent – he) and his caregiver (the child – she). Both are
altruistic, can transfer money to each other, and can provide the parent with long-term
care, the parent by purchasing formal care on the market and the child by providing
unpaid informal care. The Nash equilibrium can take three di¤erent forms as regards
money transfers: the parent (resp. the child) makes a money transfer to the child (resp.
the parent) if he (resp. she) is su¢ ciently richer than his child (resp. than her parent),
otherwise there is no money transfer. Money transfers are thus used by players to keep
the distribution of the family wealth and long-term care e¤orts within a frame that
is « acceptable » from the players’points of view, but which can be very unfair from a
regulator’s point of view. Analyzing how the Nash equilibrium is modi…ed with marginal
variations of the parameters yields surprising …ndings: the case can arise where a parent
would rather a regulator taxed his income ex ante to enrich his child, or where a parent
eligible for a lump sum public allowance would rather it was paid to his child instead of to
himself. We show that the Nash equilibrium is generally not Pareto-e¢ cient, except when
the child makes a money transfer to her parent. Not all Pareto-e¢ cient allocations can
be decentralized through an ex ante public system of taxation/subsidies of long-term care
e¤orts and ex ante lump sum transfers. To achieve Pareto-e¢ cient allocations that can
be decentralized, the regulator should subsidize informal care to reduce its opportunity
cost.

Keywords
long-term care, family transfers, altruism, Nash equilibrium, Pareto-efficient allocations, second-best allocations, taxation/subsidies of long-term care.

Classification JEL
D1, D6, H21, H31, I1

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