Optimal choice of health and retirement in a life cycle model

Michael Kuhn1, Stephan Wrzaczek2, Gustav Feichtinger2, Alexia Prskawetz 2
1Austrian Academy of Sciences, Vienna Institute of Demography, Austria, 2Vienna University of Technology, Institute of Mathematical Methods in Economics, Austria

Over the last years health and pension systems have come under increasing pressure due to rising life expectancy. There is an intensive debate about the level of health expenditures in particular for higher age groups and about the retirement age. We propose a multi-stage optimal control life-cycle model that focuses on both optimal health expenditure and retirement age and the interplay between them. By investing in health the force of mortality can be reduced. When the individual retires, wage payments are replaced by pension payment over the remaining lifespan. The pension payment depends on the length of the working period and the discounted average remaining life expectancy. In particular we distinguish between a pay as you go and a fully funded pension system as well as the social optimum. The individual has the possibility to save, where negative values are allowed over intermediate intervals of the life span (at the end of the life horizon we assume a no ponzi condition). Therefore an individual can only spend the amount of money that he/she earns in his life. As interest payment we assume an actuarially fair life insurance with the restriction that it is equal to the average mortality rate. We derive explicit expressions for the time paths of consumption and health expenditures. For the optimal retirement age no explicit expression is available, but only an implicit one. All of them have interesting economic interpretations.

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