Computing Crude Birth Rates from Total Fertility Rate

Table of Contents

Age-Specific Fertility Rate (ASFR)

Time in Years Until Population Is Halved Given TFR = 1

Earlier I Computed a Halving Period that Is Too Long

_{}

where B_{i}
= the number of live births of women in the *i*-th age
cohort of child bearing age and W_{i}
= the number of women in the *i*-th
age cohort of women of childbearing age.

“The Total Fertility Rate or TFR is the most powerful and useful measure of fertility in understanding a population. It represents the number of children the typical woman in that population will have over her childbearing years, based on 2 assumptions:

1. Women will have the same birth rates over their lifetimes as women in different age cohorts in that same population have had in that year; and

2. Women will survive through their childbearing years.”

– http://www.geo.hunter.cuny.edu/~imiyares/fertility.htm

_{}

where h
is the number of years in an age cohort, n = the number of age
cohorts of women of childbearing age, B_{i} = the
number of live births
of women in the *i*-th
age cohort
of child bearing age, and W_{i} = the number of
women in the *i*-th
age cohort of women of childbearing
age.

_{}

where
B = number of live births in a given year and B_{i}
= the number of
live births of women in the *i*-th
age cohort of childbearing age.

_{}

where P is the total number of people in the population.

No
doubt the terms 'pyramidal population distribution' and
'cylindrical population distribution' are sufficiently
evocative. The US has a cylindrical population distribution,
which is to say that each age cohort is
approximately the same size. In a mature society that does
not tolerate Die-Off,
every woman who reaches child
bearing age lives at least until she exceeds childbearing
age. Thus, we
may replace W_{i} by W_{avg} in
the denominator of the equation
for TFR:

_{}

or

_{}

Thus,

_{}

Table 1. Population, Female Populations by age cohorts, and Live Births by age cohorts, 2001.

Total population 284,796,887 4,025,933

Females 10-14 10,185,198 7,781

Females 15-19 9,843,981 445,944

Females 20-24 9,619,230 1,021,627

Females 25-29 9,333,209 1,058,265

Females 30-34 10,260,525 942,697

Females 35-39 11,138,324 451,723

Females 40-44 11,477,432 92,813

Females 45-49 10,544,119 5,083

__
__

Source: National Vital Statistics Reports, Vol. 51, No. 4, February 6, 2003, Tables 2, 11.

B = 4.026 million

P = 285 million

W_{avg}
= 10.3 million

h = 5

TFR = 2.033

Peter D. Johnson of the US Census Bureau, International Programs Center, did a simulation using the International Data Base program of which he is an author. He writes, “This shows the U.S. population dropping to half the 2005 population by about 2128 if the TFR drops to 1.0 immediately or by 2142 if the TFR drops to 1.0 by 2025. In contrast, the world population would be halved by 2079 if the TFR were immediately cut to 1.0 or by 2094 if the TFR dropped to 1.0 by 2025. The reason the U.S. takes longer is because of the assumed migration which offsets some of the losses due to the fertility declines.” He sent the following figures along with tables of data and many more figures in Excel files:

**Note.**
I was looking for a
best-case solution to the problem of Die-Off in the United States due
to Peak
Oil given that the entire population becomes rational
overnight. This was
found to be rather a long time, therefore I neglected the plateau that
would
exist in many other countries, especially in the Third World due to a
pyramidal
population distribution of age cohorts. I assumed that after
forty years
no further death of women of childbearing age occurred. The
same techniques
that I applied to the United States in this exercise were applied to
the world
population of 6.4 billion. Unfortunately, I left the crude
death rate and
the relative size of the cohorts of women of childbearing age constant,
which
is true only when the size of the population is nearly constant, i.e.,
for TFR
about two. The following shows the steps to the wrong answer:

P_{2}
= P_{1} exp{rN} = P_{1} exp{(b
– d) N}.

Let

_{}

and

_{}

Then, for TFR = 1,

ln ρ = (b-d) N = (α TFR – d)N = ( 0.0072281 – 0.0083) N = -0.0010719 N,

where I assumed that the proportion of women of childbearing age in the general population will not change. Then, for ρ = 0.5, I obtained

_{}

By
computing the crude birth rate, b, from minimum and maximum values
for W_{avg}, I bracketed the correct solution by
390.8 years < N_{1/2}
< 744.1. Next, I used the value of TFR =
2.033 to adjust the
published value of b = 0.0141 to obtain b = 0.0141 / 2.033, b
– d = r =
0.0141/2.033 – 0.0083 = -0.00136 and N_{1/2}
= 508 years. I
claimed this was a better value for N_{1/2},
however it pointed to the
same course of action (and it’s true even if the halving time
is seventy-five
years only); namely, we must reduce our use of high-grade energy to
prevent
Die-Off. No matter what else we think, this means abandoning
the capitalist-style
market economy and its associated modes of production and distribution
of goods
and services as the energetic overhead and other requirements are too
great. I have indicated why this is true in “Energy
in a
Natural Economy” as discussed in “__The
Proposition that Conservation Is a Bad Thing as an Example of
Reductio ad
Absurdum__”
and I resolve to do a better job of proving it is true in the coming
new year.

Houston, Texas

December 24, 2004

Revised January 1, 2005