Germán Rodríguez
Demographic Methods Princeton University

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 the stable equivalent age distribution, as illustrated in Box 7.2 in the textbook.

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
----------------------------------------------------------------------------------------
> L <- c(4.66740, 4.63097, 4.58518, 4.53206, 4.46912, 4.39135, 4.28969)

> m <- c(0.00567, 0.06627, 0.11204, 0.07889, 0.05075, 0.01590, 0.00610)

> a <- c(15, 20, 25, 30, 35, 40, 45) + 2.5

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     
----------------------------------------------------------------------------------------
> nrr <- sum(L * m)

> nrr
[1] 1.527414

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. The first thing we need is a little function f(r) to compute a discrete approximation to Lotka's integral (equation 7.10b in the text). We then use Coale's method, which approximates the first derivative f'(r) as minus the product of the function and the true mean age of childbearing in the stable population A, which of course is not known, so we use an estimate. The code below starts from r = log(NRR)/A with A = 27.

. capture mata mata drop f()

. capture mata mata drop coale()

. mata
------------------------------------------------- mata (type end to exit) --------------
:         function f (real scalar r, real vector a, real vector L, real vector m) {
>                 return(sum(exp(-r:* a) :* L :* m))
>         }

:         function coale(nrr, a, L, m) {
>                 r = log(nrr)/27
>                 delta = 1
>                 while (delta > 1e-8) {
>                         r0 = r
>                         f0 = f(r0, a, L, m)
>                         r = r0 + (f0 - 1)/27
>                         delta = abs(r - r0)
>                         printf(" %10.8f\n", r)
>                 }
>                 return(r)
>         }

:         coale(nrr, a, L, m)
 0.01414625
 0.01425317
 0.01424322
 0.01424413
 0.01424405
 0.01424406
  .0142440581

: end     
----------------------------------------------------------------------------------------
> f <- function(r, a, L, m) {
+   sum(exp(-r * a) * L * m)
+ }

> coale <- function(nrr, a, L, m) {
+   r <- log(nrr)/27
+   delta <- 1
+   while(delta > 1e-8) {
+       r0 <- r
+       f0 <- f(r0, a, L, m)
+       r <- r0 + (f0 - 1)/27
+               delta <- abs(r - r0)
+       cat(r,"\n")     
+   }
+   r
+ }

> coale(nrr, a, L, m)
0.01414625 
0.01425317 
0.01424322 
0.01424413 
0.01424405 
0.01424406 
[1] 0.01424406

After three iterations we agree with the textbook and after six the rate of growth changes by less than 1e-8.

An alternative is to use Newton's method by calculating the exact first derivative, which I do below by writing a short function g(r). With the same starting value

. capture mata mata drop g()

. capture mata mata drop newton()

. mata
------------------------------------------------- mata (type end to exit) --------------
:         function g(r, a, L, m) {
>                 return( sum(a :* exp(-r :* a) :* L :* m) )
>         }

:         function newton(nrr, a, L, m) {
>                 r = log(nrr)/27
>                 delta = 1
>                 while(delta > 1e-8) {
>                         r0 = r
>                         f0 = f(r0, a, L, m)
>                         g0 = g(r0, a, L, m)                      
>                         r = r0 + (f0 - 1)/g0
>                         delta = abs(r - r0)
>                         A = g0/f0
>                         printf("%10.8f  %10.4f\n", r, A)  
>                  }
>                  return(r)
>         }

:         r = newton(nrr, a, L, m)
0.01421160     29.4198
0.01424404     29.4737
0.01424406     29.4725
0.01424406     29.4725

:         r
  .0142440575

: end
----------------------------------------------------------------------------------------
> g <- function(r, a, L, m) {
+   sum(a * exp(-r * a) * L * m)
+ }

> newton <- function(nrr, a, L, m) {
+   r <- log(nrr)/27
+   delta <- 1
+   while(delta > 1e-8) {
+     r0 <- r
+     f0 <- f(r0, a, L, m)
+     g0 <- g(r0, a, L, m)  
+     r <- r0 + (f0 - 1)/g0
+     delta <- abs(r - r0)
+     A <- g0/f0
+     cat(r,A,"\n")  
+   }
+   data.frame(r=r, A=A)
+ }

> newton(nrr, a, L, m)
0.0142116 29.41982 
0.01424404 29.47367 
0.01424406 29.47248 
0.01424406 29.47248 
           r        A
1 0.01424406 29.47248

Newton converges a bit quicker, in only four iterations, and as a bonus we get the mean age of childbearing in the stable population, which is 29.47.

Lotka used a quadratic expansion to obtain an approximate solution that does not require iteration.

The Population of the U.S.

Let us now turn to the example in Box 7.2, which involves the female population of the U.S. in 1991. The data include the actual age distribution as well as the person-years lived and the maternity function. These are available in a text file called prestonBox72.dat in the datasets section.

. infile age ca L m using ///
>         http://data.princeton.edu/eco572/datasets/prestonBox72.dat, clear
(18 observations read)

. list

     +-------------------------------+
     | age      ca         L       m |
     |-------------------------------|
  1. |   0   .0726   4.95804       0 |
  2. |   5   .0689   4.95002       0 |
  3. |  10   .0667   4.94603   .0007 |
  4. |  15   .0648   4.93804   .0303 |
  5. |  20   .0729   4.92552   .0566 |
     |-------------------------------|
  6. |  25   .0799   4.91138   .0578 |
  7. |  30   .0861   4.89356   .0388 |
  8. |  35   .0801   4.86941   .0157 |
  9. |  40   .0735   4.83577   .0027 |
 10. |  45   .0556   4.78475   .0001 |
     |-------------------------------|
 11. |  50   .0464   4.70374       0 |
 12. |  55   .0421   4.57712       0 |
 13. |  60   .0436   4.38502       0 |
 14. |  65   .0429   4.10756       0 |
 15. |  70   .0365    3.7199       0 |
     |-------------------------------|
 16. |  75   .0294   3.19192       0 |
 17. |  80   .0203   2.49203       0 |
 18. |  85   .0176   2.73044       0 |
     +-------------------------------+
> box72 <- read.table("http://data.princeton.edu/eco572/datasets/prestonBox72.dat",
+   header=FALSE)

> names(box72) <- c("age", "ca", "L", "m")

> box72
   age     ca       L      m
1    0 0.0726 4.95804 0.0000
2    5 0.0689 4.95002 0.0000
3   10 0.0667 4.94603 0.0007
4   15 0.0648 4.93804 0.0303
5   20 0.0729 4.92552 0.0566
6   25 0.0799 4.91138 0.0578
7   30 0.0861 4.89356 0.0388
8   35 0.0801 4.86941 0.0157
9   40 0.0735 4.83577 0.0027
10  45 0.0556 4.78475 0.0001
11  50 0.0464 4.70374 0.0000
12  55 0.0421 4.57712 0.0000
13  60 0.0436 4.38502 0.0000
14  65 0.0429 4.10756 0.0000
15  70 0.0365 3.71990 0.0000
16  75 0.0294 3.19192 0.0000
17  80 0.0203 2.49203 0.0000
18  85 0.0176 2.73044 0.0000

Eigenanalysis of the Leslie Matrix

The first step will be to construct a Leslie matrix and compute the first eigenvalue and eigenvector. Make sure you first cut and paste the function to compute a Leslie matrix from the population projection log.

. mata:
------------------------------------------------- mata (type end to exit) --------------
:         L = st_data(.,"L")

:         m = st_data(.,"m")

:         M = Leslie(L, m)

:         values = J(0, 0, .)

:         vectors = J(0, 0, .)

:         eigensystem(M, vectors, values)

:         values[1]
  .999167947

:         log(values[1])/5
  -.00016648

:         sa = Re(vectors[,1]/sum(vectors[,1]))

: end
----------------------------------------------------------------------------------------
> M <- Leslie(box72$L, box72$m)

> e <- eigen(M)

> Re(e$values[1])
[1] 0.999168

> log(Re(e$values[1])/5)
[1] -1.61027

> sa <- Re(e$vectors[,1])

> sa <- sa/sum(sa)

The intrinsic rate of growth of the U.S. in 1991 was r = -0.0001665. If the observed fertility and mortality patterns were to prevail the population of the U.S. would eventually decline about 0.017% per year.

Estimating r using Coale's Method

Let us now calculate the rate using Coale's method to solve Lotka's equation iteratively. We start by computing the NRR, which as you would expect is less than one. We also need the midpoints of the age groups. For the last group we use 6.79, life expectancy at age 85, as in the textbook

. mata
------------------------------------------------- mata (type end to exit) --------------
:         nrr = sum( m :* L )

:         nrr             
  .9956019291

:         a = range(0,85,5) :+ 2.5

:         a[18] = 85 + 6.79

:         coale(nrr, a, L, m)
 -0.00016642
 -0.00016648
 -0.00016648
  -.0001664831

: end
----------------------------------------------------------------------------------------
> nrr <- sum(box72$m * box72$L)

> box72 <- mutate(box72, a = ifelse(age < 85, age + 2.5, 85 + 6.79))

> coale(nrr, box72$a, box72$L, box72$m)
-0.0001664199 
-0.0001664812 
-0.0001664824 
[1] -0.0001664824

As you can see the method settles on r = -0.0001665, in agreement with the earlier result. Let us try using the exact first derivative

. mata
------------------------------------------------- mata (type end to exit) --------------
:         r = newton(nrr, a, L, m)
-0.00016648     26.4788
-0.00016648     26.4789

: end
----------------------------------------------------------------------------------------
> n <- newton(nrr, box72$a, box72$L, box72$m)
-0.0001664826 26.47876 
-0.0001664824 26.47888 

> r <- n$r

The method converges very quickly and gives a mean age of childbearing in the stable population of 26.479. Note that there is a typo in the textbook, which reports r as -.00018 on page 150.

Obtaining the Equivalent Age Distribution

The final step will be to calculate the stable equivalent age distribution. This is a simple function of r and the person-years lived. (It also involves the birth rate, but that is also a function of r and person-years lived.) Below we compute b and then the stable age distribution, which we compare to the eigenvector

. mata:
------------------------------------------------- mata (type end to exit) --------------
:         b = 1/sum( exp(-r*a) :* L )

:         b
  .0125842618

:         saa = b * exp(-r*a) :* L        

:         st_store(.,st_addvar("float","sa"), sa)

:         st_store(.,st_addvar("float","saa"), saa)

: end
----------------------------------------------------------------------------------------

. list age ca sa saa      

     +-----------------------------------+
     | age      ca         sa        saa |
     |-----------------------------------|
  1. |   0   .0726   .0624188   .0624193 |
  2. |   5   .0689   .0623697   .0623702 |
  3. |  10   .0667   .0623714   .0623718 |
  4. |  15   .0648   .0623225   .0623229 |
  5. |  20   .0729   .0622162   .0622167 |
     |-----------------------------------|
  6. |  25   .0799   .0620893   .0620897 |
  7. |  30   .0861   .0619155   .0619159 |
  8. |  35   .0801   .0616613   .0616617 |
  9. |  40   .0735   .0612863   .0612867 |
 10. |  45   .0556   .0606902   .0606906 |
     |-----------------------------------|
 11. |  50   .0464   .0597123   .0597127 |
 12. |  55   .0421   .0581533   .0581537 |
 13. |  60   .0436    .055759   .0557594 |
 14. |  65   .0429   .0522744   .0522748 |
 15. |  70   .0365   .0473803   .0473806 |
     |-----------------------------------|
 16. |  75   .0294   .0406893   .0406896 |
 17. |  80   .0203   .0317938   .0317941 |
 18. |  85   .0176   .0348964   .0348897 |
     +-----------------------------------+
> b <- 1/sum(exp(-r * box72$a) * box72$L)

> b
[1] 0.01258426

> saa <- b * exp(-r * box72$a) * box72$L

> cbind(box72$ca, sa, saa)
                     sa        saa
 [1,] 0.0726 0.06241882 0.06241925
 [2,] 0.0689 0.06236975 0.06237018
 [3,] 0.0667 0.06237137 0.06237180
 [4,] 0.0648 0.06232247 0.06232290
 [5,] 0.0729 0.06221623 0.06221665
 [6,] 0.0799 0.06208928 0.06208971
 [7,] 0.0861 0.06191552 0.06191594
 [8,] 0.0801 0.06166127 0.06166169
 [9,] 0.0735 0.06128628 0.06128670
[10,] 0.0556 0.06069017 0.06069059
[11,] 0.0464 0.05971232 0.05971273
[12,] 0.0421 0.05815331 0.05815371
[13,] 0.0436 0.05575903 0.05575942
[14,] 0.0429 0.05227440 0.05227476
[15,] 0.0365 0.04738031 0.04738064
[16,] 0.0294 0.04068929 0.04068958
[17,] 0.0203 0.03179383 0.03179406
[18,] 0.0176 0.03489637 0.03488968

As you can see the do calculations of the stable equivalent age distribution give essentially the same result. Watcher's textbook explains why this is so.

Let us now plot the current and stable age distributions using the midpoints of the age groups

. mata st_store(.,st_addvar("float","am"), a)

. line ca sa am, lpat(dash solid) ///
>    xtitle("age") ytitle(Prop in 5-year age group) ///
>    title(Actual and Stable Equivalent Age Distributions) ///
>    subtitle(U.S. females 1991) ///
>    legend(order(1 "Actual" 2 "Stable") col(1) ring(0) pos(1))

. graph export stablepop.png, width(500) replace
(file stablepop.png written in PNG format)

Stable

> box72$sa <- saa

> library(tidyr)

> g72 <-  select(box72, age=a, current=ca, stable=sa) %>% 
+   gather(key = "distribution", value= "proportion", -age)

> ggplot(g72, aes(age, proportion, color=distribution)) + geom_line() +
+   ggtitle("U. S. Females 1991")

> ggsave("stablepopr.png", width = 500/72, height = 400/72, dpi = 72)

Stable The graph reproduces Figure 7.4 in the textbook, showing how the 1991 age distribution has proportionately more people at younger ages (up to 40-44) and fewer at older ages (45-49 and up).

A Preview of Momentum

The youthful U.S. agre structure means that, at 1991 rates, the U.S. population would continue to grow for a while before it starts to decline, so it has positive population momentum. How long would it grow and what size would it reach? The following code snipet answers both questions

. mata:
------------------------------------------------- mata (type end to exit) --------------
:         delta = 1

:         c0 = st_data(.,"ca")  // current age distribution

:         n=0

:         first = 1

:         while (delta > 1e-8) {
>                  c1 = M * c0
>                  r1 = log(sum(c1)/sum(c0))/5
>                  delta = abs(r1-r)
>                  n++
>                  // print when pop first declines
>                  if(r1 < 0 & first) {
>                          n, sum(c1), r1, delta
>                          first = 0
>                  }
>                  c0 = c1
>          }
                  1              2              3              4
    +-------------------------------------------------------------+
  1 |             9    1.138400948   -.0002235479    .0000570647  |
    +-------------------------------------------------------------+

:         n, sum(c1), r1
                  1              2              3
    +----------------------------------------------+
  1 |            53      1.0844633   -.0001664872  |
    +----------------------------------------------+

: end
----------------------------------------------------------------------------------------
> delta <- 1

> c0 <- box72$ca

> n <- 0

> first <- TRUE

> while (delta > 1e-8) {
+   c1 <- M %*% c0
+   r1 <- log(sum(c1)/sum(c0))/5
+   delta <- abs(r1 - r)
+   n <- n + 1
+   if(r1 < 0 & first) {
+       cat(n, sum(c1), r1, delta, "\n")
+       first <- FALSE
+   }
+   c0 <- c1
+ }
9 1.138401 -0.0002235473 5.706484e-05 

> data.frame(n, size = sum(c1), r1)
   n     size            r1
1 53 1.084464 -0.0001664865

We see that the population grows for 9 projection periods (45 years), reaching a size 13.84% larger than in 1991, and then heads for extinction, with the growth rate reaching -0.0001665 (to a close approximation) after 56 projection periods (280 years), at which time it is still 8.45% larger than at the start. (So extinction is not imminent!)

Question: On our way to the stable age distribution we computed the intrinsic birth rate. What's the intrinsic death rate?