Some extensions of the keyfitz momentum relationship

A stable population, such that the total birthrateB(t) =Boerot, is abruptly altered by modifying the age-specific birth rate,m(x). The survivor function remains unaltered. The modified population ultimately settles down to a stable behavior, such thatB(t) =B1er1t. It is shown thatB1/B0 = (R0 −R1)/[(r0 −r1)R0Z1], whereR0,R1 are the net reproduction rates before and after the change, and expected age giving birth for the stable population after the change. The age structure and transients resulting from the change are also described. The effect of an abrupt change in the survivor functionl(x) is also investigated for the simple case where the change is caused by alteringl(x) toe−λxl(x). It is shown that the above ratio becomes, whereN refers to the numbers in the population,k =r0 + λ, andg(x) =m(x)l(x), the value before the change. A measure for the reproductive worth of the population is also established.