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


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.

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Fig. 1


  1. Chakraborty (2004) introduced this specification to analyze the relationship between endogenous lifetime and economic growth. De la Croix and Ponthière (2010) used it to address the optimal accumulation of capital under endogenous longevity. Felder and Mayrhofer (2011) applied this specification in a one-period of life model to derive optimal investment in longevity.

  2. Beside the basic model with consumption in the first- and the second period of life, De la Croix et al. (2013) incorporated exogenous old-age labor and endogenous labor supply. Schweizer (1996, p. 210) criticized the quest for extended versions of the serendipity theorem: “It might be fun to derive serendipity type and Henry George type results for general settings. The more tedious task, however, which consists of ensuring a maximum rather than a minimum of well-being should always be kept in mind.”

  3. See the Appendix for the derivation of the second-order conditions.

  4. In the industrialized world, the median public share of total health spending is 71.5 % (OECD 2014).


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

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Correspondence to Stefan Felder.

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Responsible editor: Alessandro Cigno



If we assume that the population growth rate is exogenous, three first-order conditions remain for the planner’s problem, Eqs. (3), (4), and (12) in the model with costly survival. Substituting the resource constraint (11) for c, the Hessian matrix of the second-order derivatives associated with this problem is

$$ H=\left[\begin{array}{ccc}\hfill \frac{\partial^2\mathrm{E}\mathrm{U}}{\partial {d}^2}\hfill & \hfill \frac{\partial^2\mathrm{E}\mathrm{U}}{\partial d\partial k}\hfill & \hfill \frac{\partial^2\mathrm{E}\mathrm{U}}{\partial d\partial m}\hfill \\ {}\hfill \frac{\partial^2\mathrm{E}\mathrm{U}}{\partial k\partial d}\hfill & \hfill \frac{\partial^2\mathrm{E}\mathrm{U}}{\partial {k}^2}\hfill & \hfill \frac{\partial^2\mathrm{E}\mathrm{U}}{\partial k\partial m}\hfill \\ {}\hfill \frac{\partial^2\mathrm{E}\mathrm{U}}{\partial m\partial d}\hfill & \hfill \frac{\partial^2\mathrm{E}\mathrm{U}}{\partial m\partial k}\hfill & \hfill \frac{\partial^2\mathrm{E}\mathrm{U}}{\partial {m}^2}\hfill \end{array}\right] $$

Hence, \( \left[\begin{array}{ccc}\hfill {\left[\frac{\pi (m)}{1+n}\right]}^2u^{\prime\prime }(c)+\pi (m)u^{\prime\prime }(d)\hfill & \hfill -u^{\prime\prime }(c)\left(f^{\prime }(k)-n\right)\frac{\pi (m)}{1+n}\hfill & \hfill \begin{array}{l}u^{\prime\prime }(c)\frac{\pi (m)}{{\left(1+n\right)}^2}\left[1+n+\pi^{\prime }(m)d\right]\\ {}-u^{\prime }(c)\frac{\pi^{\prime }(m)}{1+n}+u^{\prime }(d)\pi^{\prime }(m)\end{array}\hfill \\ {}\hfill -u^{\prime\prime }(c)\left(f^{\prime }(k)-n\right)\frac{\pi (m)}{1+n}\hfill & \hfill \begin{array}{l}u^{\prime\prime }(c){\left[\left(f\prime (k)-n\right)\right]}^2\\ {}+u^{\prime }(c)f^{\prime\prime }(k)\end{array}\hfill & \hfill -\frac{u^{\prime\prime }(c)}{1+n}\left[1+n+\pi^{\prime }(m)d\right]\left({f}^{\prime }(k)-n\right)\hfill \\ {}\hfill \begin{array}{l}u^{\prime\prime }(c)\frac{\pi (m)}{{\left(1+n\right)}^2}\left[1+n+\pi^{\prime }(m)d\right]\\ {}-u^{\prime }(c)\frac{\pi^{\prime }(m)}{1+n}+{u}^{\prime }(d){\pi}^{\prime }(m)\end{array}\hfill & \hfill -\frac{u^{\prime\prime }(c)}{1+n}\left(f^{\prime }(k)-n\right)\left[1+n+\pi^{\prime }(m)d\right]\hfill & \hfill \begin{array}{l}\frac{u^{\prime\prime }(c)}{{\left(1+n\right)}^2}{\left[1+n+\pi \prime (m)d\right]}^2\\ {}-u^{\prime }(c)\frac{\pi^{\prime\prime }(m)d}{1+n}+\pi^{\prime\prime }(m)u(d)\end{array}\hfill \end{array}\right] \) Substituting the first-order conditions yields

$$ \left[\begin{array}{ccc}\hfill {\left[\frac{\pi (m)}{1+n}\right]}^2u^{\prime\prime }(c)+\pi (m)u^{\prime\prime }(d)\hfill & \hfill 0\hfill & \hfill u^{\prime\prime }(c)\frac{\pi (m)}{{\left(1+n\right)}^2}\frac{\pi^{\prime }(m)u(d)}{u^{\prime }(d)}\hfill \\ {}\hfill 0\hfill & \hfill u^{\prime }(c)f^{\prime \prime }(k)\hfill & \hfill 0\hfill \\ {}\hfill u^{\prime\prime }(c)\frac{\pi (m)}{{\left(1+n\right)}^2}\frac{\pi^{\prime }(m)u(d)}{u^{\prime }(d)}\hfill & \hfill 0\hfill & \hfill \frac{u^{\prime\prime }(c)}{{\left(1+n\right)}^2}{\left[\frac{\pi \prime (m)u(d)}{u\prime (d)}\right]}^2+u^{\prime }(c)\frac{\pi^{\prime\prime }(m)}{\pi^{\prime }(m)}\hfill \end{array}\right] $$

A sufficient condition for (c, d, k, m) to be a maximum is that the Hessian matrix is negative definite. The second-order conditions (19)–(21) in the text are the three conditions concerning the determinants of the three submatrices of size 1 × 1, 2 × 2, and 3 × 3 that guarantee the negative definiteness of H.

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Felder, S. “How powerful is demography? The serendipity theorem revisited” comment on De la Croix et al. (2012). J Popul Econ 29, 957–967 (2016).

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  • Longevity
  • Health expenditure
  • Overlapping generations
  • Value of a statistical life

JEL classification

  • E13
  • H42
  • H75
  • I18
  • J17