# “How powerful is demography? The serendipity theorem revisited” comment on De la Croix et al. (2012)

## Abstract

Samuelson’s (Int Econ Rev 16(3):531-538, 1975) serendipity theorem states that the “goldenest golden rule” steady-state equilibrium can be obtained by a competitive two-period overlapping generation economy with capital accumulation, provided that the optimal growth rate prevails. De la Croix et al. (J Popul Econ 25:899-922, 2012) extended the scope of the theorem by showing that it also holds for risky lifetime. With this note, we introduce medical expenditure as a determinant of the probability of surviving to old age to prove the theorem. The original as well as all extended versions of the serendipity theorem, however, fail to prove that second-order conditions are satisfied in general. Still, unlike De la Croix et al. (J Popul Econ 25:899-922, 2012), we can exclude the existence of corner solutions where the probability of reaching old age is zero or one. The zero survival probability case becomes irrelevant if the option to randomize between death and life utility is taken into account. Survival with certainty is ruled out if the marginal cost of survival is increasing. Hence, the optimal survival probability represents an interior solution. Furthermore, we show for the optimal survival probability that the value of a statistical life is positive and equal to its marginal cost.

## Keywords

Longevity Health expenditure Overlapping generations Value of a statistical life## JEL classification

E13 H42 H75 I18 J17## Notes

### Acknowledgments

I wish to thank Friedrich Breyer for most helpful comments and Denis Bieri for technical assistance. Comments by two referees are also gratefully acknowledged.

## Compliance with ethical standards

## Conflict of interest

The author declares that he has no conflict of interest.

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