Calculating r using an iterative method (cont.)

A worked example using empirical fertility and mortality schedules (cont.)

We can then use iterative methods to get a more precise estimate of the growth rate. It is clear that a slightly lower value of r is needed to balance the equation, and it would be possible to keep guessing values of the growth rate to move towards a sum of 1.0. But Coale’s iteration formula for a subsequent value of r allows us to refine our estimate of the growth rate more efficiently. In this formula, r1 is the previous estimate of r and x1 the previous sum of Lotka’s equation. The NRR in this population is 2.319.

r= r 1 .ln(NRR) (ln( NRR )− x 1 +1)

So in this case, our next growth rate estimate will be:

r= 0.0311.ln(2.319) (ln( 2.319 )−0.93155+1) =0.0288

Once again we insert the new value of the growth rate in to Lotka’s equation to see if we have found the correct growth rate for the given schedules of fx and Lx. If the equation still does not converge to 1, we repeat the process until it does:

Age group Mid-point Female age-specific fertility rate Life table person years Iterations
5fx.5Lx.exp{-ra}
x - x+4 a 5fx/2.05 5Lx 1 2 3 4
15-19 17.5 0.03234 4.76625 0.08937 0.09312 0.09334 0.09334
20-24 22.5 0.10356 4.74050 0.24360 0.25679 0.25758 0.25760
25-29 27.5 0.12429 4.70725 0.24844 0.26499 0.26599 0.26601
30-34 32.5 0.10400 4.66750 0.17640 0.19037 0.19122 0.19124
35-39 37.5 0.07741 4.61900 0.11120 0.12142 0.12204 0.12206
40-44 42.5 0.03795 4.55550 0.04601 0.05083 0.05113 0.05114
45-49 47.5 0.01624 4.46675 0.01652 0.01847 0.01859 0.01859
NRR = 2.319 Lotka sum 0.93155 0.99598 0.99989 1.00000
G ≈ 27 r estimate 0.0311 0.0288 0.0287 0.0287

After 4 iterations, we can see that Lotka’s equation has produced a value of 1. This means we have found the correct growth rate for the given schedules of fx and Lx.