DEMOG 110 (Fall 2022) Problem set 5 solutions Andrea Mirand

DEMOG 110 (Fall 2022)
Problem set 5 solutions

Andrea Miranda-Gonz ́alez
October 2022

1 Question 1
From chapter 4 question 2 of Wachter 2014:
“Table 4.6 shows estimates of cohort age-specific fertility rates and person-years lived for the cohort of
Swedish women born in 1800. The radix is 1,000. Find the cohort NRR, TFR, and GRR. How close is the
GRR to the NRR?”

We also need a value for the fraction female at birth and for the initial cohort size. Since the question doesn’t
give us any information on the number of babies by sex, we cannot calculate the ff ab. Instead, we use the
default value of 0.4886.
ff ab = 0.4886
ℓ0 = 1000
We can expand the table to include more columns.

The GRR is about 1.793 times larger than the NRR. There is almost a full child (0.997) difference between
the GRR and NRR.
Interpretation of results of TFR, GRR and NRR
• TFR: if we assume that a woman lives through her complete reproductive span and experiences the
age-specific fertility rates throughout her entire lifespan, she should have approximately 4.614 children.
• GRR: if we assume that a woman lives through her complete reproductive span and experiences the
age-specific fertility rates throughout her entire lifespan, she should have approximately 2.254 female
children.
• NRR: taking into account the mortality of women in a cohort, the average number of daughters in a
cohort is 1.257.

2 Question 2
From chapter 4 question 4 of Wachter 2014: “Table 4.7 shows women by parity at ages 45 to 50 in the whole
sample from the 2004 DHS in Malawi discussed in Section 4.4. Calculate counts of women at and above
each parity along with the values of PPR(j) for all j.”
Solution:
The counts of women at and above a certain parity mean all the women that have at least j children.

Table 2: Table 4.7 Women by completed parity, Malawi 2004
Source: Demographic and Health Survey (DHS), Malawi, 2004
For example the number of women that are at parity 10 and above are w(10+) = w(10) + w(11) +
w(12) + w(13) + w(14) + w(15). The number of women that are parity 11 and above are w(11+) =
w(11) + w(12) + w(13) + w(14) + w(15). Notice the w(10+) contains w(11+). This means that you can add
w(j) from bottom to top to obtain the number of women at parity j and above.
This value comes in handy in order to calculate the P P R(j). The formula in our book say that:

3 Question 3
From chapter 4 question 5 of Wachter 2014:
“From Table 4.7, calculate estimates of the cohort TFR and GRR for this cohort from Malawi. What is the
mean completed parity for women at and above parity 8? By how much would we underestimate the TFR
if all women at and above parity 8 were assigned to parity 8?”
Solution:
• There is an important nuance in this question and it has to do with the source of the data and who
the respondents are. In section 4.4 (Wachter 2014), the author notes that “Sometimes data come in
the form of a distribution of children ever born. Such data make possible a third way of calculating
the NRR”. This data concern the completed parity of women in a cohort, which means that we follow
the women of a given cohort until they die. Therefore the counts take into account cohort mortality
(in other words they don’t assume that women survive through their reproductive lifespan).
• However, we cannot always follow a cohort of women and sometimes we only have the “women who
have survived to an age like 50 and then report numbers of children ever born”. This is common in
surveys such as the Demographic and Health Survey (DHS). Remember, at a given time you can only
survey people that are alive (survivors up to a certain age).
• The data distinction is important because it will help us decide what we can calculate with the data
provided. For this question, the table comes from the DHS, which surveys women between the ages
of 45 and 50. These are the women from a specific cohort that survived their childbearing years. By
definition, the TFR and GRR assume that women live through the end of their reproductive lifespan.
Therefore with survey data we can calculate the TFR and GRR.
• We cannot calculate the NRR with survey data because we don’t actually know about the completed
parity of women that died before age 45 and 50. In this sense, the data does not reflect completed
parity for all women in a cohort.
What is the cohort TFR and GRR?
To calculate the cohort TFR, we use the ‘mean completed parity for these surviving women’, which refers
to the sum of total children birthed by all the women of the cohort, divided by the number of initial people in the cohort.

What is the mean completed parity for women at and above parity 8?
From the previous question, we know that the mean completed parity for all women is the number of children

4 Question 4
True or false: doubling the fertility rate at all ages doubles the TFR. Give a short explanation for your
answer. Solution:

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