Introduced by Samuelson (1975), the Serendipity Theorem states that
the competitive economy will converge towards the optimum
steady-state provided the optimum population growth rate is
imposed. This paper aims at exploring whether the Serendipity
Theorem still holds in an economy with risky lifetime. We show
that, under general conditions, including a perfect annuity market
with actuarially fair return, imposing the optimum fertility rate
and the optimum survival rate leads the competitive economy to the
optimum steady-state. That Extended Serendipity Theorem is also
shown to hold in economies where old adults work some fraction of
the old-age, whatever the retirement age is fixed or chosen by the
agents.