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notes Stable Populations

# notes Stable Populations - Eco 572 Research methods in...

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Eco 572: Research methods in Demography Stable Populations In this unit we do some stable population calculations, including determination of the intrinsic growth rate, as illustrated in Box 7.1, and computing he stable equivalent age distribution, as illustrated in Box 7.2 in the textbook. I used Mata throughout, which I think leads to clearer code, but I need it only for the eigenanalysis in the third section; otherwise all calculations can be done in plain Stata, as shown in the alternative version. Box 7.1. The Population of Egypt We showed in the previous unit how to calculate r from the first eigenvalue of the Leslie Matrix. We now use the Egyptian example in Box 7.1 to illustrate Coale's method. We start by entering person-years lived for ages 15-19 to 45- 49, the maternity function at those ages, and the midpoints of the age groups. . mata: ------------------------------------------------- mata (type end to exit) ------------------------------------------- : L = (4.66740, 4.63097, 4.58518, 4.53206, 4.46912, 4.39135, 4.28969) : m = (0.00567, 0.06627, 0.11204, 0.07889, 0.05075, 0.01590, 0.00610) : a = (15,20,25,30,35,40,45) :+ 2.5 : end --------------------------------------------------------------------------------------------------------------------- The Net Reproduction Ratio The Net Reproduction Ratio NRR is easily computed as the sum of the products of the survival ratios and the maternity function . mata: ------------------------------------------------- mata (type end to exit) ------------------------------------------- : nrr = sum( L :* m ) : nrr 1.527413734 : end --------------------------------------------------------------------------------------------------------------------- The NRR is 1.527 daughters per woman, in agreement with the textbook. Coale's Method for Estimating r Next we solve Lotka's equation. Coale's method follows from a Taylor expansion of Lotka's integral (equation 7.10b in the text) which I'll call f(r), and its first derivative f'(r)=-f(r)A(r) where A is the mean age of childbearing written as a function of r. If r0 is a trial value and r the true value or solution, write f(r0) = f(r) + (r0-r)f'(r) and note that f(r)=1 and f'(r)=-A where A is the true mean age of childbearing in the stable population. The next trial value would then be r = r0 + (f(r0)-1)/A. As the textbook notes, we don't know A until we know r, so we use an approximation, in this example 27. For a starting value we use r = log(NRR)/A. We write a one-line function to compute a discrete approximation to Lotka's integral given a trial value of r . We then use log(NRR)/27 as a trial value and compare the result to unity to determine a new trial value. The code below does three iterations . mata: ------------------------------------------------- mata (type end to exit) ------------------------------------------- : scalar f(scalar r, vector a, vector L, vector m) { > return( sum(exp(-r * a) :* L :* m ) ) > } http://data.princeton.edu/eco572/stablepop.html (1 of 7) [2/12/2008 11:07:34 AM]

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