Exercise 9.23. Consider the continuous time perpetual youth model of Section 9.8. (1) Show that given L(0) = 1, the initial size of a cohort born at the time τ ≥ 0 is exp ((n − ν) τ ).

(2) Show that the probability that an individual born at the time τ is alive at time t ≥ τ is exp (−ν (t − τ )). (3) Derive equation (9.40). (4) Show that this equation would not apply at any finite time if the economy starts at t = 0 with an arbitrary age distribution.