Lectures No 1 and 2
September 9 and September 11, 2003
Measures
Conventions:
i. I will use the symbol “*” to denote multiplication; thus x*y is x times y
ii. I will use the symbol “Sum(Rj)” to denote summation of the Rj’’s over all j’s
iii. I will refer to somebody aged x as a person whose last birthday was the xth (e.g. he/she is in
the age group x,x+1, that is, aged x but not quite x+1)
iv. The symbol “~”denotes approximation
a. What=s a rate?
{Events in an interval of time} / {Population exposed in same interval}
events============>count of events of interest
population exposed===>person years lived =population in midperiod * width of period
If interval of time is t=1
CBR=births in year t / midyear population in year t
CDR=deaths in year t / midyear population in year t
NMR=net migrants in year t / midyear population in year t
What would the expressions look like if interval is of width 5?
Note: The NMR is the difference between the Crude Immigration Rate and the Crude
Outmigration Rate
b. rates and probabilities: the issue of boundary values
A probability is ratio of events to number of experiments;
Thus, a probability of dying in one year would be deaths / population exposed
As a consequence, a probability can vary between 0 and 1 whereas a rate can
vary between 0 and infinity
Example:
10 people born at the same time live to be 19 and go on to celebrate their 20th birthday to
Dictator Island; they are caught by the military, accused of terrorism, and shot at the
same time a few days before celebrating their 20th birthday.
a. the probability of dying between 19 and 20 is 1 and zero elsewhere;
b. the death rate in the interval 19 to 20- is 0 but becomes infinity at exact age 20;
c. Rate(crude) of increase and natural rate of increase
r = CBR – CDR
R = r + NMR
this the natural rate of increase
this is the rate of increase
d. Doubling times
If a population experiences a constant rate of increase r , its doubling time or the time it takes for
its size to double, is given by Td~.69/r…..Do the math: what does it mean for the absolute size of
the population of India, for example, that its r is about .02? When r=0 then the doubling time is
infinity. In the 1960’s Kenya was growing with r~.04. What doubling times does this imply?
e. CDR and age specific mortality rates
What is an age specific mortality rate?
Mx = Dx / Px
Where Dx are death in a year in the age group x,x+1 and Px is the midyear population in the age
group x,x+1
What does it look like? The biology of it all…………………..
FIGURE 1: A MORTALITY CURVE
What is a CDR in terms of the Mx’s?
CDR =
Sum of Dx / Sum (Px)
Sum [(Dx / Px) * (Px / Sum(Px))]
Sum [Mx * Cx]
Where Cx is the proportion of the population in the age group x,x+1
The problem is that since CDR depends on Cx as well as Mx, comparisons of CDR may reflect
effects of age distribution. In general older populations have HIGHER CDR even if their MX’s
values are lower throughout.
Solution 1: Standardization
We use a standard set of values Cx’s to recalculate all CDR being compared.
Solution 2: Life table
We convert Mx into S(x)=====> the life table
An important relation:
if interval is small enough the following is true: Mx is constant
if so, Px=exp(-Mx) is the probability of surviving from age x to x+1
S(x)=1*P0*P1*……Px-1
Start with N(0)=100,000 and proceed to calculate N(x)
Life expectancy at birth, Eo, is average age at death of the population or expected number of
years to be lived after birth
Life expectancy at age x, Ex, is number of years expected to live after age x for survivors to age
x
FIGURE 2: THE SURVIVAL CURVE
e. Variability of mortality: Models of mortality
modelling a la Brass
modelling a la Coale-Demeny
f. CBR and age specific fertility rates
CBR decomposed:
Bx / P=(Bx/Wx)*(Wx/35W15)*(35W15/W)(W/P))
Where:
Bx are births to women aged x,x+1
Wx are women aged x,x+1
50W15 are women in the age group 15-49
W is total female population
P is total population
=Fx * Cx * 35C15 * w
where Fx is an age specific fertility rate
Cx is the age distribution of females
35C15 is the proportion of all females who are in the age group 15-49
w is the fraction of all population who are females
What does Fx look like? The biology of it all.
FIGURE 3 : A FERTILITY CURVE
We can generate different measures by using only the Fx’s
Total Fertility Rate=TFR=Sum(Fx)
Gross Reproduction Rate=GRR=TFR*.45
If we add consideration that a recently born female has a chance of surviving to age x which is
S(x), then her NET fertility at that age will be Fx*S(x)
Net Reproduction Rate=NRR=Sum(Fx *S(x))
g. Models of marital fertility
There is a fundamental distinction between marital and non marital fertility: marital fertility
takes place within unions, however defined; non-marital fertility takes place outside unions
What is nx, natural fertility?
Answer No 1 based on intervals only
Suppose fertile period goers from age15 to 50 exactly
Suppose mean birth interval is composed of:
8 months of wait for postpartum amenorrhea
5 months of wait for conception
4 months of wait due to pregnancy losses
9 months of wait for normal pregnancy
Then the average number of births would be:
35*12 / 26 or ~16
Answer No 2 based on Henry’s idea
Natural fertility (represented by age specific fertility rates labeled nx) occurs when the behavior
does not change as a function of past reproductive performance
What are deviations from nx?
level: different societies may have different fecundity levels. This affects the height of n(x). Say
that when fecundity is lower it can be represented by a constant K<1. And when it is higher by a
value of K>1. So that the new value of nx would be nx’=K*nx
Age pattern: women may control fertility to stop childbearing. In this case there is a departure
from nx that depends on age according to a pattern vx and an intensity m. Thus, new nx can be
represented as
nx’= nx * exp(-vx*m)
Combining both we get the following expression for any fertility rate for married women,
gx=K*nx*exp(vx*m)
If a fraction Gx of those women aged x are married, then
fx= Gx*gx
FIGURE 4 : CURVE OF NATURAL FERTILITY
FIGURES 5 AND 6 : VARIETIES OF
FERTILITY CURVES
h. Standardized measures of fertility: Princeton fertility study
Note: all sums below are between ages 15 and 50 exactly
Let Fx be fertility rate for all women aged x,x+1;
Let gx be fertility rate for married women aged x,x+1;
Let hx be the fertility rate for the Hutterite population in age x,x+1
Let Wx and =WMx denote all women aged x, x+1 and married women aged x,x+1
If= Sum (Fx*Wx) / Sum Wx
Ig= Sum (gx*WMx) / Sum (hx* WMx)
Im= Sum (WMx * hx)/ Sum (Wx*hx)
The contours of fertility: iso-fertility curves.
The notion of an indifference curve
FIGURE 7: A GRAPH OF ISO –R CURVES
Historical strategies along the iso curves
i. Age distributions, Cx
pyramides, the role of past history to effect age distributions
varieties of age distributions
FIGURE 8 : AGE PYRAMIDE FROM FRANCE
FIGURE 9 : AGE DISTRIBUTIONS FOR 4 COUNTRIES
j. How are age distributions determined?
Case 1: stationary
Note: Below the notation F(x) where F(.) is a function of x signifies the exact value of F at x.
closed to migration
constant Mortality or S(x)
constant number of births every year
We get N(x)=B*S(x)
We get C(x)=S(x) / Sum(S(x))
Case 2: stable
closed to migration
constant Mortality or S(x)
exponential number of births: B(t)=B* exp(r*t)
N(x)=B(t-x)*S(x)
B(t-x)=Bo*exp(r*(t-x))
N=Sum of N(x)
C(x)=N(x) / N
C(x)=CBR*S(x)*exp(-r*x)
Stationary population is a special case
C(x)=(1/e)*S(x)
a unique relation: NRR=exp(r*T) where T ~ mean age at childbearing
j. Important relations:
a. changes in mortality affect the age distribution very little:
when mortality is high, a decline in mortality leads to a YOUNGER population
when mortality is low, a decline in mortality leads to an OLDER population
b. changes in fertility have a whooping effect on age distribution:
invariably a decline in fertility translates into an OLDER population
FIGURE 10: GRAPHS OF AGE DISTRIBUTIONS
c. momentum of population growth: the fraction by which a population would
continue to grow before reaching r=0 if NRR went down to 1.00 in a single
instant of time
M=( CBR* Eo / r * T) *(NRR-1/NRR)