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How to measure population growth

How to measure population growth. Crude rates b or CBR Crude Birth rate It expresses the number of births per 1 (100, 1000) inhabitants in the year t d or CDR Crude Death rate It expresses the number of deaths per 1 (100, 1000) inhabitants in the year t

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How to measure population growth

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  1. Howtomeasure • populationgrowth

  2. Crude rates • b or CBR Crude Birth rate • It expresses the number of births per 1 (100, 1000) inhabitants in the year t • d or CDR Crude Death rate • It expresses the number of deaths per 1 (100, 1000) inhabitants in the year t • We can also calculate crude immigration and emigration rate. • Generally speaking, crude rates express the number of events per inhabitants (or per 100,1000 inhabitants) in the specified year.

  3. Population momentum (a simple exercise) Pop. year t2 (fertility decreases - Pop. increases) W0-14 500 W15-49 1500 W50- 1000 Risk of having a child during t2 = 0.08 Expected births = 0.08*1500=120 • Pop. year t1 • W0-14 1000 • W15-49 1000 • W50- 1000 • Risk of having a child during t1 = 0.1 • Expected births = 0.1*1000=100 Pop. year t3(Fertility is constant -Population reduces) W0-14 500 W15-49 500 W50- 2000 Risk of having a child during t3 = 0.08 Expected births= 0.08*500=40 Population momentum is strictly linked to the age structure!

  4. Population Balance Equation • The change in the population size in a given time interval (0-t) is the sum of Natural increase (births – deaths) and Net migration(immigrants – emigrants): In almost all countries, natural increase is the most important component of overall population change over time. When we cannot (or we do not want to) consider migrations, we refer to a closed population where :

  5. Population Balance Equation • World population • pop (1.1.2006)= 6.555 billion • pop(1.1.2007)=6.625 billion • Births(2006)=129 million • Deaths (2006)= 59 million • Pop. change= 70 million • US • Population change 1990-1991 (number in thousands) • ΔP=Pt - P0 = 250878-248168 = • = B - D + I - E = 4179-2162+853-160=2710 • ΔP gives the extent of variation but it depends on the population dimension. • How can we compare the growth between different populations?

  6. The exponential growth rate • The easiest way: consider the ratio between the variation in a certain period of time and the initial population. The growth rate is: • Where ΔP=Pt-P0is the variation and P0 is the initial pop. • In order to have an annual growth rate we have: We are assuming a constantgrowthover time, i.e. a linearpopulation growth. Every year the growth is ΔP/t . Let us consider a (closed) population with P0= 100. During the following t=40 years we have registered 10 deaths and 50 births. Therefore ΔP=50-10=40 . The annual variation is 40/40=+1 individual and the annual growth rate is:

  7. The exponential growth rate • Under the hp of linear growth, the variation does not depend on the current population size. • In our example, the annual increase is equal to one individual when the current pop size is 100 as well as when the pop size is 140. • In other words, only the initial population ( P0 )contributes to the growth whereas the new arrivals never give their contribution to the growth (think to a constant interest rate repeatedly applied to an initial capital). • This hp is plausible only in the short period (one year). • In the long run, new individuals that enter in the population will progressively contribute to the variation. Thus, under the linear hp, the longer the period of time (t) the higher the bias in the growth rate estimation.

  8. The exponential growth rate • It is more likely that the annual variation (ΔP/t) increases as the population grows. • We may assume that the relative variation remains constant. • Let us suppose that the variation is equal to 2% per year (constant rate): the number of new arrivals will increase year by year. We are assuming that the population is growing exponentially. In the continuous time (changes are allowed in each point in time) we have where e is a numeric constant (Eulero’s number) equal to 2.718

  9. The exponential growth rate • From the previous expression we obtain the exponential growth rate (continuous time): Note: this is calculated like the growth of an investment with compound interest (in the continuous time). In our example: P0=100 Pt=140 t=40 The exponential growth rate is NOTE: the number of new arrivals increases every year (in the linear hp the number of new arrivals is constant).

  10. The exponential growth rate • Using a a mathematical model, the exponential growth rate equation gives the opportunity to extrapolate data (useful for missing data in the past and for projections in the future). However, use it carefully! • Examples of application: • 1. • In the 1852 the USA population was projected to the 2000 using an extrapolation based on exponential model. • Initial population 31 million (census data) • Projected population (2000) 703 million • 2000 US Census counted 281 million (a strong growth due to demographic transition but not so strong as predicted)

  11. The exponential growth rate • 2. World population (seen before): • 1 million years ago 125,000 • 10,000 b.C. 10 million • 1 A.D. 250 million • 1650 A.D. 500 million • Exponential growth rate (in the long run): • - Period: 1 million y. a. – 10,000 b.C. • - Period: 10,000 b.C. – 1 A.D. Starting from the Neolithic the growth rate is 70 times higher.

  12. The exponential growth rate The value remains similar to the previous period (1.3 times higher). • - Period: 1 A.D. – 1650 • Note that the population has doubled in that period (1650 years to double with an annual rate equal to 0.042%) • - Following periods: • 1650-1930 r = 0.49% annual • 1930-2000 r = 1.57% annual BE CAREFUL! The Hp is that the growth rate remains constant throughout the period. This is a strong approximation. When data are available, we should calculate the growth rate for shorter periods especially in the more recent past (no more than 50 years in the last centuries).

  13. The exponential growth rate • 3. • Doubling time • The hypothetical numberof years until the population will double if r remains constant • Halving time • The hypothetical numberof years until the population will halve if r remains constant • . (where ln2=0.693 and ln 0.5= - 0.693)

  14. The exponential growth rate • 3. (continued) For example, in the period 1 – 1650 AD we see that the population doubled with a growth rate of 0.00042. We may check that the doubling time is consistent t=0.693/0.00042=1650 years Doubling time (t*) according to r values Rule of 70: doubling time is approximately E.g. at the growth rate of 2.8% per year, the population of Uganda would double in 70/2.8=24 years if that growth rate remained unchanged.

  15. Crude growth rate • When we have information not only on total population but also on births, deaths and migrations, we can compute the growth rate using the population equation • r = b - d+ i - e(for a closed population: r = b - d) and we call it crude growth rate. It is used only for short periods (usually one year). It can be demonstrated that for a single year this rate corresponds to the exponential growth rate. World pop., 1950-2000 (rates per thousand)

  16. The growth rate • To sum up, • - if we have only population size at two (or more) points in time •  exponential growth rate • - if we also have the annual number of births, deaths, (migrations) •  crude growth rate • In any case, • r population growth rate • It expresses the pace of the growth of a population measuring the annual mean variation (number of individuals added or subtracted in a single year per 1000 individuals) • For example, if for a certain year r=-3 per 1000, it means that in this year the population has reduced by 3 individuals every 1000. • In contemporary populations r varies between -1% and +3% .

  17. World population at the beginning of 2009: 6,769,614,784 Compute the world population growth rate in the 2009.

  18. Exercises • I.1 • The total amount of a certain population at 31.12.1991 and at 31.12.2001 is respectively 5320 and 5624 inhabitants. • a. Compute the annual growth rate in the decade considering an exponential growth. • b. Compute the doubling time supposing that the growth rate remains constant in the considered period.

  19. Exercises - Solutions • I.1 • a. • We know that: • P31.12.1991=5320 • P31.12.2001=5624 • Under the hypothesis of exponential growth we have Pt=P0ertand then • b. • We obtain the doubling time t* considering that Pt*=2P0. Therefore: • A quick way to compute the doubling time with an acceptable degree of approximation is consider the ratio between 70 and the annual growth rate (expressed as a percentage). In this case, considering that r=0.056 %, the doubling time is 70/0.56=125 years.

  20. Exercises I.2 For a certain population the following data are reported: Compute the population size at 31.12.2000 supposing that the exponential growth rate remains constant over time. 1.3 Given the population size of Turin (census years 1961-1991), compare the changes occurred in the two periods of time (assuming an exponential growth model).

  21. Exercises - Solutions I.2 P31.12.1995=5300 P31.12.2005=4900 P31.12.2000.? Firstly, we compute r (constant in the decade 31.12.1995 - 31.12.2005). Then, 1.3

  22. Appendix • Exponential function • Logarithmic function (e base) (Number of Nepero) Properties of logarithms

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