To get a real idea of the nature of population reproduction, indicators are needed that do not depend on the sex and age structure. In the early 1930s. German demographer, economist, statistician R. Kuchinsky (1876-1947) and domestic scientist, demographer, health care organizer G.A. Batkis (1895-1960) used indicators that give a clear picture of the state of the number of the new and old generation in the years adjacent to the years of the population censuses, helping to determine the extent to which the living population has prepared a replacement for itself:

total fertility rate;

gross reproduction rate;

net reproduction rate.

The total fertility rate shows the number of children born on average by one woman for the entire fertile period of her life (i.e. from 15 to 49 years inclusive). It is calculated like this:

where nx is the age-specific fertility rate for women aged x years.

The calculation can also be performed for five-year intervals:

and for 10 year olds:

An example of calculating the total fertility rate is given in table. one.

Table 1. Calculation of the total fertility rate for the rural population of the Novosibirsk region, 1999

As follows from the table. 1, for their entire fertile period, every 1000 rural women in the Novosibirsk region will give birth to 1404 (1403.5) children, i.e. 1.414 on average per woman, or roughly 140 children per 100 women.

The total fertility rate as an indicator of population reproduction is not without its drawbacks. Thus, he does not take into account: first, that the reproduction of the new generation can be characterized by the number of girls that each woman leaves behind; secondly, that some children die before reaching the age of their mother at the time of their birth, leaving behind no offspring or leaving fewer children in comparison with their peers who have happily survived to the end of the childbearing period.

The first drawback can be eliminated using the gross reproduction rate R b, calculated by the formula

where d is the proportion of girls born.

For the example shown in table. 1, and for d - 0.488

R b = 1.4035 0.488 = 0.6849.

Consequently, every 1000 women leaves behind 685 girls (684.9), i.e. even simple reproduction is not carried out in the rural population of the region.

The advantage of the gross coefficient is that its value is not influenced by the composition of the population by sex and that it takes into account the age composition of women of fertile age. However, it does not take into account the mortality rate of women of fertile age.

For the most accurate characteristics of population reproduction, the net coefficient is used. In the statistical literature, it is called pure or purified. It shows the number of girls that each woman leaves behind on average, taking into account the fact that some of them will not live up to the age of their mother at the time of their birth.

However, if each of their women of reproductive age gives birth to R daughters on average, this does not mean that the number of daughters 'generation will be R times greater or less than the number of mothers' generation. After all, not all of these daughters will live to the age at which their mothers were at the time of birth. And not all daughters will survive to the end of the reproductive period. This is especially true in countries with a high mortality rate, where up to half of the newborn girls may not survive until the beginning of the reproductive period, as was the case, for example, in Russia before the First World War. In our time, of course, this is no longer the case (in 2004, more than 98% of newborn girls survived to the beginning of the reproductive period), but in any case, an indicator is needed that also takes into account mortality. Taking into account the assumption of zero mortality until the end of the reproductive period, the gross reproduction rate of the population has practically not been published or used recently. An indicator that also takes into account mortality is the net reproduction rate of the population, or otherwise the Boeck-Kuchinski coefficient, proposed by the German statistician and demographer G.F.R. Byeck. Otherwise, it is called the net reproduction rate of the population. It is equal to the average number of girls born in a woman's lifetime and surviving to the end of the reproductive period, given the birth and death rates.

The following formulas are used to calculate the net coefficient Rn:

a) for one-year age groups:

where n x - age coefficients for women in the age group X years; d is the proportion of girls born;

The average number of women living in the stationary population of life tables in the age range from X to X + 1;

b) for five-year age groups:

where - age-specific fertility rates for women in the age group from X to X + 4;

Average number of living women from life tables in the age range from X to X + 4 (+ +1 + +2 + +3 + +4);

c) for ten-year age groups:

where - age-specific fertility rates for women in the age group from X to X + 9;

The average number of women living in the stationary population of persons of survival in the age range from x to x + 9.

Example. The number of women in the stationary population of the Novosibirsk region (according to life tables) and age-specific fertility rates are known:

Let's calculate the net reproduction rate. Let's define the "expected" number of children.

With the proportion of girls born d = 0.488 Rn = 135 5490.488:

100,000 = 0.66148, or rounded 0.662.

Consequently, every 1000 rural women leaves behind only 662 girls. This confirms the initial conclusion that a mode of narrowed reproduction has been established in this population.

The advantage of the net coefficient lies in the fact that it takes into account the birth rate in certain age groups of women at the time of compilation of life tables, and when calculating it, the mortality rate of the population, the probability of surviving to the next age group are taken into account. In statistical practice, the following scale for assessing the net reproduction rate is adopted: when Rn = 1.0, simple reproduction occurs; for Rn> 1.0 - extended, for Rn< 1,0 -- суженное.

B.S. Yastremsky established the relationship between the total fertility rate, fertility rate (special fertility rate, fertility rate) and reproduction rates (Tables 2 and 3).

Table 2. Relationship between fertility rates

Table 3. The relationship between fertility and reproduction rates

Therefore, the border between narrowed and simple reproduction lies between the meanings:

· Special birth rate from 100 to 150 ‰;

· Gross reproduction rate from 0.86 to 1.29 ‰;

· The total fertility rate from 15 to 22 ‰.

The net reproduction rate can be calculated not only for the female population, but also for the male population using the same method. In this case, it shows how many boys each man leaves behind, taking into account the fact that some of them will not live up to the age of their father at the time of their birth.

To calculate the net reproduction rate of the male population by one-year groups, the following formula can be used:

where are the age coefficients of the birth of children in families for men of the age group x years,

The number of living men in the stationary population of life tables in the age range from X years to X + 1;

d M is the proportion of boys born.

The calculation is carried out in a similar way for five- and ten-year age groups.

Table 4. Initial data for calculating the reproduction rates of the male and female population of the region, people

Note. Age groups: for women - 15-49 years old, for men - 18-55 years old.

Let's calculate the number of births per 1000 people of the population (n x) as (N x: S x 1000).

Age group

45 and older

Average

Hence the total fertility rate according to the formula:

51,000 for women:

=(78,3 + 226,7 + 193,2 + 106,2 + 36,3 + 8,9 + 1,6)5:1000 = 3,26;

for men:

+ (23,0 + 234,3 + 231,2 + 146,6 + 68,3 + 18,2 + 5,7)5:1000 = 3,64,

those. each woman leaves an average of 3.26 children for the entire fertile period of her life, a man - 3.64.

The gross reproduction rate of the population is calculated by the formula R b =:

3,260,488 = 1,591;

3,640,512 = 1,864,

those. each woman left on average 1,591 girls, men - 1,864 boys.

To go to the definition of the net coefficient, we calculate the "expected" number of children:: 1000, for example,

for women: 78.3485 117: 1000 = 37 985;

for men: 23.0487 370: 1000 = 11210, etc.

Net reproduction rate:

for women formula

for men formula

Consequently, every 1000 women on average leaves behind 1529 girls, taking into account the fact that some of them will not live to the age of their mother at the time of their birth, and every 1000 men - 1724 boys, provided that some of them will not survive to the age of their father at the moment of their birth. their birth. The net coefficient of the male population is higher than the net coefficient of the female population by 0.196 points, or 12.8%.

In the second half of the XX century. in the world there was a tendency towards a decrease in all three indicators of population reproduction, and for economically developed countries it crossed the boundaries of simple reproduction (Fig. 1).

Rice. one.

The first turning point in the modern demographic history of Russia is 1964, when the fall in the net reproduction rate of the Russian population crossed the line of generational replacement. In the same year, the mortality curve began to creep upward, which, in the end, led to the shameful modern level of life expectancy of Russians.

Period X is a characteristic resonant surge caused by politics and the environment of the 80s: a slow, jerky rise, a small upper plateau and an accelerating collapse well below the point of initial growth. It is noteworthy that the collapse of the population reproduction rate began long before the arrival of the "criminal liberal government" and a sharp deterioration in the socio-economic situation of the Soviet people.

Period Y - is divided into two political eras: the Yeltsin era, when uncertainty grew and the socio-economic situation of the majority of the country's population worsened; and the Putin era - when certainty grew, the vertical of power was strengthened, the socio-economic situation improved, the optimism of the voting majority multiplied.

The graph clearly shows the growth of the curve since 1999, the post-default year: the pre-active demographic policy is still 8 years old.

According to UN forecasts, by the period 2010-2014. the regions with narrowed population reproduction will include Europe Abroad, Asia Abroad, Australia and Oceania. The highest level of the net ratio will remain in Africa. And 109 women will leave behind 109 girls in America.

In Russia, the process of narrowed reproduction is deepening (see Table 5.)

Table 5. Dynamics of the net reproduction rate of the population in the Russian Federation in 1960 - 2000.

The narrowed reproduction of the urban population began by the end of the 1950s, the rural population - from 1993.

In 2000, every 1000 women of fertile age left 529 girls in cities, and 704 in rural areas.

According to the Demographic Yearbook, the total fertility rate for the period from 1991 to 2000 varied across the CIS countries from 1.10 in Ukraine to 4.09 in Turkmenistan. In Europe in 1999, the Czech Republic had the lowest level of the indicator - 1.12, and France - 1.77. In Asia for 1995-2000. the highest level was reached by Iran - 5.30 and Saudi Arabia - 5.80, the lowest - Japan - 1.39; China had 1.80, India had 3.40. In Africa, the total fertility rate reached 3.81 in Algeria, 3.74 in Egypt, and 3.25 in South Africa (1995-2000). In America for 1995-2000. Canada had the lowest level of indicator - 1.64, the highest - Mexico - 2.75; in the USA -2.02; in Australia - 1.80 (1996), in New Zealand - 1.97 (1997).

However, if each of the women of reproductive age gives birth on average R daughters, this does not mean that the number of daughters' generation will be R times more or less than the size of the generation of mothers. After all, not all of these daughters will live to the age at which their mothers were at the time of birth. And not all daughters will survive to the end of the reproductive period. This is especially true in countries with a high mortality rate, where half of the newborn girls may not survive until the beginning of the reproductive period, as was the case, for example, in Russia before the First World War 2. In our time, of course, this is no longer the case (in 1997, almost 98% of newborn girls survived to the beginning of the reproductive period, however, in any case), an indicator is needed that also takes into account mortality. Taking into account the assumption of zero mortality until the end of the reproductive period, the gross reproduction rate of the population has practically not been published or used recently.

An indicator that also takes into account mortality is net reproduction rate of the population, or otherwise, Beck-Kuchinski coefficient . Otherwise, it is called the net reproduction rate of the population. It is equal to the average number of girls born in a woman's lifetime and surviving to the end of the reproductive period, given the birth and death rates. The net reproduction rate of the population is calculated using the following approximate formula (for data for five-year age groups):

where all designations are the same as in the formula for the gross coefficient, a 5 L x f and l 0 - respectively, the number of people living in the age interval (x + 5) years from the female mortality table. The formula for calculating the net reproduction rate of the population uses the number of people living in the age interval (x + n) years from the female mortality table, and not the function of survival, that is, not the number of survivors before it begins (l x), because it is an approximate formula. In rigorous demographic analysis and mathematical applications of demography, it is the survival function that is used 1 (x).

Despite the somewhat "threatening" appearance, this formula is quite simple and allows without any special difficulties, especially using the appropriate software, for example, Excel spreadsheets, to calculate the value of the net reproduction rate of the population. In addition, many programs have been developed that allow you to reduce the calculation of the net coefficient to a simple input of the initial data. For example, the IPC of the U.S. Bureau of the Census has developed a PAS (Population Spreadsheets Analysis) spreadsheet system, one of which (SP) is based on data on the values of age-specific fertility rates and the number of people living in the age interval (x + n) years calculates gross and net reproduction rates, as well as the true rate of natural increase and generation length, which will be discussed below 3.

Table 7.1 is an example of calculating the age-specific fertility rate, gross and net reproduction rates of the population, in which the above software is not used. Using this example, as well as a similar example given in V.A. Borisov 4, you can easily learn how to calculate all the main indicators of population reproduction. But, of course, it is desirable to have at least some kind of computing equipment, it is best, of course, to use Excel.

The calculation was made according to the following step-by-step procedure:

Step 1. In column 2 we enter the values of the age-specific fertility rates (5 ASFR X, taken in this case from the Demographic Yearbook of the Russian Federation for 1999 (p. 155 **).

Step 2. We calculate the total fertility rate (TFR). For this, the number in the rows of column 2 is divided by 1000 in order to express the age-specific fertility rates in relative shares of 1 (in other words, we bring these values to 1 woman of a conditional generation). We enter the obtained quotients in column 3. The sum of these numbers, multiplied by 5, gives us the value of the total fertility rate equal to 1.2415 (highlighted bold italic). This, to within the third decimal place, coincides with the official data of the State Statistics Committee of the Russian Federation (1.242. WITH. 90).

Step 3. We calculate the gross reproduction rate (TO), or the number of daughters a woman has born in her lifetime. To do this, we multiply the data in column 3 row by row by the proportion of girls among newborns (D). In this case, its average value for the period 1960-1998 was taken, equal to 0.487172971301046. The sum of the numbers in column 4 multiplied by 5 gives the gross reproduction rate equal to 0.6048. The same result can be obtained by simply multiplying the total fertility rate by the proportion of girls among newborns (1.2415 0.487 ... = 0.6048).

Step 4. In column 5, we enter the values \ u200b \ u200bof the numbers living at each age interval (x + 5 years (x = 15, 20, ..., 45) from the mortality table for the female population of Russia for 1998. In column 6, these numbers are reduced to relative fractions of a unit by dividing them by the root of the mortality table (in this case, by 10,000). An alternative way is the averaging of two adjacent values of the numbers surviving to the beginning of each age interval from 15 to 50 years from the mortality table for the female population for 1998 (p. 188). Multiplying the obtained averages by 5, we determine the necessary for calculating the number of people living in each age interval.

Step 5. We calculate the net reproduction rate. To do this, we multiply the data in column 4 row by row by the numbers in column 6. Summing up column 7, we obtain the value of the net reproduction rate equal to 0.583. This value differs only by 0.002 from that officially published by the State Statistics Committee of the Russian Federation (0.585, p. 114 of the Demographic Yearbook for 1999).

The net reproduction rate is calculated for a conditional generation. As a measure of replacement of the maternal generation by the generation of daughters, it is valid only for the so-called stable population, in which the reproduction regime does not change, i.e. fertility and mortality. The size of such a population changes (i.e. increases or decreases) in R 0 once in a while T, called the average generation length.

Calculation of indicators of reproduction of the population of Russia for 1998 5

Table 7.1

Generation length

Generation length is the average time interval separating generations. It is equal to the average age of a mother at the birth of daughters who live at least to the age at which their mothers were at the time of their birth.

To calculate the length of a generation, an approximate formula can be used, which is given in many demography textbooks 6:

where all designations are the same as in the previous formula. As can be seen from the formula, the desired generation length is obtained as the arithmetic mean of the ages of mothers at the birth of daughters (in this case, the middle of the corresponding age interval is used.) the moment of their birth. Please note that calculating generation length is exactly the same as calculating the average age at childbirth, which we did in the chapter on fertility. The only difference is in the weights used (when calculating the average age at the birth of a child, as you remember, age-specific fertility rates were used as weights) and in the fact that in this case we are not talking about all children born, but only about daughters, and only those of them who live at least to the age of their mother at birth.

Let's return now again to table. 7.1 and let's take the last, sixth step.

Step 6. We calculate the length of a generation, or the average age of a mother at the birth of daughters who live at least to the age at which their mothers were at the time of their birth. For this, the numbers in the rows of column 7 are multiplied by the middle of each age interval (column 8) and we enter them in column 9. The resulting products represent the number of person-years lived by all daughters born by 1 woman of a conditional generation in a given age interval and living at least to the age of their mother at the time of their birth. Summing up these products, we get the numerator of the above formula for calculating the length of a generation, approximately equal to 14.8709. This number is the number of person-years lived by all daughters born by 1 woman of a conditional generation throughout her life and living at least to the age of their mother at the time of their birth. Dividing this last value by the number of all such daughters, that is, by the net reproduction rate of the population (0.5859), we obtain the required length of the female generation in Russia in 1998. For the data we have chosen, it is equal to 25.38232512, or 25 , 38 years old.

True rate of natural growth As mentioned above, the net reproduction rate of the population (R 0) shows that the size of a stable population, corresponding to the real one, with the given general fertility and mortality rates, which are assumed to be unchanged, changes (i.e., increases or decreases) in R 0 times per time T, that is, for the length of a generation. Taking this into account and accepting the hypothesis of exponential growth (decline) of the population, we can obtain the following relationship between the net coefficient and the length of a generation. This relationship is derived from the following equation: Р Т = Р () R 0 = Р 0 - e r T (remember Chapter 3, that section of it, which talks about the rates of growth and population growth):

In the theory of a stable population, r in these expressions is called the true rate of natural population growth (or A. Lotka's coefficient). This coefficient is the root of the so-called integral equation of population reproduction, or Lotka 7 equation. It is widely used in mathematical applications of demography, in particular in the theory of a stable population. However, we do not consider this equation here, since this topic is beyond the scope of this tutorial. Those interested are referred to the Course of Demography, ed. AND I. Boyarsky (M, 1985, pp. 90-91 and 103-118), as well as to the corresponding articles of the Demographic Encyclopedic Dictionary (Moscow, 1985) and the Encyclopedic Dictionary “Population” (M, 1994). For a very close approximate solution of the Lotka equation with respect to the true coefficient and generation length, as well as the computational procedure, see: Shryock H.S., Sigel J.S. The Methods and Materials of Demography / Condensed Edition by E.G. Stockwell. N.Y. San Francisco, London 1969. P. 316-31.8.

Lotka Alfred James (1880-1949), American biologist and demographer. [...] President of the American Population Association (1938-1939), American Statistical Association (1942) ... and fertility and mortality rates. ... For the first time he proposed a mathematical expression for the own coefficient of natural growth of a closed population with a constant order of extinction and childbearing, the algebraic expression of which was given in the work "On the true coefficient of natural growth of the population" (1925), showing the relationship of this coefficient with the net rate of population reproduction. .. Lotka studied the process of generational change, gave a modern analytical expression of the length of a generation ...

Population. Encyclopedic Dictionary. M., 1994.S. 210.

The last formula, proposed by the American demographer E. Cole, already familiar to you from the chapter on fertility, in his article “Calculation of Approximate True Ratios” 8, can be used to estimate the true rate of natural population growth, given that, as mentioned above, the length of a generation is the average the age of the mother at the birth of daughters who live at least to the age at which their mothers were at the time of their birth. In modern conditions, the length of a generation does not differ much from the average age of the mother at childbirth *. Therefore, the evaluation of the last parameter in any way allows you to approximately establish both the sign and the value of the true coefficient of natural growth.

If we now use E. Cole's formula and divide the just calculated length of the female generation by the natural logarithm of the net reproduction rate (lnO, 5859 = -0.534644249954392), then we get the true natural growth rate of the population of Russia for the conditions of 1998. This value is -0.0210636435922121, or = -2.1%.

The real value of the coefficient of natural growth of the population of Russia in 1998 was equal to -0.48%, or almost 4.4 times less in absolute value. This difference is due to the relatively high proportion of women of reproductive age in the population of Russia, which, in turn, is associated with a slight increase in the birth rate in the first half of the 1980s. last century and with the influence of previous demographic waves. The real age structure of our country is younger than the age structure corresponding to the modern parameters of fertility and mortality of a stable population. The population has accumulated some growth potential, or, more precisely, the potential for inhibition of population decline, due to which the population of our country is not decreasing as quickly as it would otherwise be.

But this situation will end very soon. Generations born during the decline in fertility, which began in the second half of the 1980s, will begin to enter reproductive age. last century and continuing to this day **. And then the potential for demographic "growth" will be exhausted, and the natural decline in the population of our country, if no measures are taken, will be even faster (in 4 -5 times faster than now). And no replacement migration, which some demographers rely on will not save our country from the horrors of depopulation.

For example, in the same 1998, the average age of a mother at childbirth, according to S.V. Zakharov, was 25.34 years old. See: Population of Russia 1999. Seventh annual demographic report / Otv. ed. A.G. Vishnevsky. M., 2000. S. 55. Goskomstat of the Russian Federation gives a value of 25.3 years (see: Demographic Yearbook of the Russian Federation 1999, p. 170).

The increase in the number of births in the past two years is nothing more than an artifact.

Although, strictly speaking, the net reproduction rate is a measure of the replacement of the maternal generation by the generation of daughters, it is usually interpreted as a characteristic of the replacement of generations in the entire population (not only women). In this case, the nature of the replacement of generations (population reproduction) is assessed in accordance with the following rule:

The clarification "after a time equal to the length of a generation" is very important. If R 0< 1, this does not mean that in the year for which the net reproduction rate is calculated, there is a decrease in the population, the absolute number of births and the total fertility rate. The population can grow for quite a long time, despite the fact that the value of the net coefficient is less than or equal to 1. This has been the case, for example, in Russia since the end of the 60s. Until 1992, the value of the net coefficient in our country all these years was less than 1, respectively, the true coefficient of natural increase was negative, and the population increased due to the potential for demographic growth accumulated in a relatively young age structure. Only when this potential was exhausted (and this happened just in 1992), the birth rate became lower than the death rate, and the population began to decline in numbers.

We can say that depopulation in Russia from hidden, latent has become explicit and open. And this did not depend at all on the specific political and socio-economic situation of the 90s. of the last century, no matter what the so-called "nationally concerned scientists" and self-proclaimed "patriots" of any color, from ultra-left to ultra-right, may say. The beginning of depopulation in our country was predetermined by the processes that took place in the population throughout the entire XX century, especially in the post-war period, when there was a sharp drop in the need for children, which caused a rapid and deep drop in the birth rate. This, in fact, is the case in all developed countries. About a third of the countries in the world have a birth rate that is less than what is necessary for simple reproduction of the population. In other words, in these countries, as in Russia, there is a latent or overt depopulation. And most of these countries are those in which the standard of living of the population is much higher than in our country.

In the previous paragraph, it was said about the birth rate necessary to ensure simple reproduction of the population. This raises the question of how to determine this birth rate. Different methods are used to answer it.

One of them was proposed by V.N. Arkhangelsky 9. The method is based on a simple comparison of the actual total fertility rate with its notional value equal to the total death rate. The ratio of the second to the first shows (in fact, this is the reciprocal of the vitality index, which was discussed at the beginning of the chapter), how many times the value of the total fertility rate should be greater in order to guarantee zero natural population growth at a given mortality rate and the available age structure:

where TFR h, TFR a, GMR, GBR- respectively, the hypothetical total fertility rate necessary to ensure simple reproduction, the actual total fertility rate, the crude death rate and the crude fertility rate.

The gross and net coefficients provide an opportunity differently, but it is also quite simple to answer this question. For this, either the ratio of the net ratio to the gross ratio, or the inverse ratio, is used.

The first ratio, that is, the ratio of the net coefficient to the gross coefficient (R0 / R), shows what is the level of potential reproduction of the population, or, in other words, how many women in each next generation replace women of the previous generation per one born girl 10.

Inverse ratio, i.e. the ratio of the gross ratio to the net ratio (R / R 0), shows how many girls a woman of a conditional generation needs to give birth to in order to ensure simple reproduction of the population. It is usually denoted by the Greek letter r:

In particular, for our example (see table 7.1):

From this it is easy to obtain the value of the total fertility rate required to ensure simple reproduction of the population. To do this, you just need to divide this expression by the proportion of girls among newborns, that is, by the secondary sex ratio:

Calculation according to the method of V.N. Arkhangelsk gives the value of the total fertility rate required to ensure simple reproduction, approximately equal to 2.04, which is much less. Apparently, this difference is reflected in the fact that the method associated with the use of gross and net coefficients gives the ratio of fertility and mortality in its pure form, and in the method of V.N. Arkhangelsky, the role of the age structure is also taken into account. It is interesting to compare the dynamics of the hypothetical total fertility rate (TFR h), calculated by these two methods, for 1996-1998.

If we use the calculations of V.A. Borisov, it turns out that the value of the hypothetical total fertility rate (TFR h), calculated by the method of V.N. Arkhangelsk, in 1996 was equal to approximately 2.05, that is, we have a decrease in two years by 0.01. The calculation by the alternative method gives for 1996 the value TFR h, equal to 2.12, which, on the contrary, is 0.01 more than 11. As you can see, the dynamics of the hypothetical total fertility rate calculated by various methods turned out to be the opposite. With mortality declining at that time, this difference can be explained both by a slight rejuvenation of the age structure of the reproductive contingent, and by an increase in the gap in the dynamics of fertility and mortality (fertility continued to fall even faster than before, and mortality also decreased slightly, but not in the same proportion ).

In Russian literature, p is sometimes called at the cost of simple reproduction. It is believed that its value characterizes the so-called. "Economy" of population reproduction, or the ratio of demographic "Costs" and "Results"."Costs" are respectively measured by the gross ratio, and "results" by the net ratio. Moreover, the lower the value of p and the closer it is to 1, the more "economical" is the reproduction of the population 12. The application of the supposedly "economic" terminology to population reproduction seems somewhat strange (it is not clear what to do with ethics here). In addition, it seems that the name of this indicator ("The price of simple reproduction"), and its interpretations in the mouths of many of our demographers are needed only in order to prove to ourselves and to our readers that the situation with reproduction in our country is far from one that could cause alarm. What, in fact, to worry about if the value of p in our country is practically the same as in the forefront countries of the West. We are, so to speak, if not ahead of the rest of the planet, then, at least in the front ranks progressive humanity.

To be involved in progress is, of course, impressive. But the question arises whether this is progress. Can you call progress an inexorable and rapid fall into the abyss of depopulation? Unfortunately, many demographers either ignore these damned questions, or refer to the negative demographic dynamics in our country, at best, conciliatory, and at worst, even believing modern demographic trends (especially the situation with the birth rate) as something quite normal.

All the indicators of population reproduction described above refer to the female population. However, in principle, similar indicators (gross and net reproduction rates, the true rate of natural increase, the length of the male generation, etc.) can be calculated for the male population, as well as for the entire population. Analysis of the reproduction of the male population in recent years has become more widespread in demography. Above, we have already discussed one of the successful examples of this kind of analysis carried out by V.N. Arkhangelsk. However, their consideration is beyond the scope of our book.

Keywords

Reproduction of the population, replacement of generations, reproduction mode, vitality index, gross coefficient, net coefficient, stable population, true coefficient of natural growth, Lotka coefficient, generation length, simple reproduction, narrowed reproduction, expanded reproduction, the price of simple reproduction.

Review questions

1. What is the relationship between the concepts of natural population growth (decline) and population reproduction?

3. What is the difference between gross and net reproduction rates?

4. What is the Lotka factor and what exactly does it mean?

5. How is the “simple reproduction price” calculated? What is the methodological role of this indicator?

2. Indicators of population reproduction: total fertility rate, gross reproduction rate, net reproduction rate

Total fertility rate;

Gross reproduction rate;

Net reproduction rate.

The total fertility rate shows the number of children born on average by one woman for the entire fertile period of her life (i.e. from 15 to 49 years inclusive). It is calculated like this:

where nx is the age-specific fertility rate for women aged x years.

The calculation can also be performed for five-year intervals:

and for 10 year olds:

An example of calculating the total fertility rate is given in table. one.

Table 1. Calculation of the total fertility rate for the rural population of the Novosibirsk region, 1999

Mother's age, years	Average age fertility rate per year,%	"Expected" number of children for the entire age interval

The first drawback can be eliminated using the gross reproduction rate R b, calculated by the formula

where d is the proportion of girls born.

For the example shown in table. 1, and for d - 0.488

R b = 1.4035 0.488 = 0.6849.

Consequently, every 1000 women leaves behind 685 girls (684.9), i.e. even simple reproduction is not carried out in the rural population of the region.

The following formulas are used to calculate the net coefficient Rn:

a) for one-year age groups:

where n x - age coefficients for women in the age group X years; d is the proportion of girls born;

The average number of women living in the stationary population of life tables in the age range from X to X + 1;

b) for five-year age groups:

R b =

where - age-specific fertility rates for women in the age group from X to X + 4;

Average number of living women from life tables in the age range from X to X + 4 (+ +1 + +2 + +3 + +4);

c) for ten-year age groups:

R b = ,

where - age-specific fertility rates for women in the age group from X to X + 9;

The average number of women living in the stationary population of persons of survival in the age range from x to x + 9.

Example. The number of women in the stationary population of the Novosibirsk region (according to life tables) and age-specific fertility rates are known:

Let's calculate the net reproduction rate. Let's define the "expected" number of children.

Age group (years)

44,3487400:1000=21592

121,5484863:1000=58911

71,7481410:1000=34517

28,8477476:1000=13751

11,1472404:1000=5244

3,2465094:1000=1488

0,1454729:1000=46

With the proportion of girls born d = 0.488 Rn = 135 5490.488:

100,000 = 0.66148, or rounded 0.662.

Consequently, every 1000 rural women leaves behind only 662 girls. This confirms the initial conclusion that a mode of narrowed reproduction has been established in this population.

B.S. Yastremsky established the relationship between the total fertility rate, fertility rate (special fertility rate, fertility rate) and reproduction rates (Tables 2 and 3).

Table 2. Relationship between fertility rates

Table 3. The relationship between fertility and reproduction rates

Therefore, the border between narrowed and simple reproduction lies between the meanings:

· Special birth rate from 100 to 150 ‰;

· Gross reproduction rate from 0.86 to 1.29 ‰;

· The total fertility rate from 15 to 22 ‰.

To calculate the net reproduction rate of the male population by one-year groups, the following formula can be used:

where are the age coefficients of the birth of children in families for men of the age group x years,

The number of living men in the stationary population of life tables in the age range from X years to X + 1;

d M is the proportion of boys born.

The calculation is carried out in a similar way for five- and ten-year age groups.

Table 4. Initial data for calculating the reproduction rates of the male and female population of the region, people

Note. Age groups: for women - 15-49 years old, for men - 18-55 years old.

Let's calculate the number of births per 1000 people of the population (n x) as (N x: S x 1000).

Age group

Women

Men

45 and older

Average

Hence the total fertility rate according to the formula:

51,000 for women:

=(78,3 + 226,7 + 193,2 + 106,2 + 36,3 + 8,9 + 1,6)5:1000 = 3,26;

for men:

+ (23,0 + 234,3 + 231,2 + 146,6 + 68,3 + 18,2 + 5,7)5:1000 = 3,64,

those. each woman leaves an average of 3.26 children for the entire fertile period of her life, a man - 3.64.

The gross reproduction rate of the population is calculated by the formula R b =:

3,260,488 = 1,591;

3,640,512 = 1,864,

those. each woman left on average 1,591 girls, men - 1,864 boys.

To go to the definition of the net coefficient, we calculate the "expected" number of children:: 1000, for example,

for women: 78.3485 117: 1000 = 37 985;

for men: 23.0487 370: 1000 = 11210, etc.

Net reproduction rate:

for women formula

R b = ): ;

for men formula

): .

Consequently, every 1000 women, on average, leaves behind 1529 girls, taking into account the fact that some of them will not live up to the age of their mother at the time of their birth, and every 1000 men - 1724 boys, provided that some of them will not live up to the age of their father at the time of their birth. birth. The net coefficient of the male population is higher than the net coefficient of the female population by 0.196 points, or 12.8%.

Rice. 1. Curve of the net coefficient for 1960-2006.

The first turning point in the modern demographic history of Russia was 1964, when the fall in the net reproduction rate of the Russian population crossed the line of generational replacement. In the same year, the mortality curve began to creep upward, which, in the end, led to the shameful modern level of life expectancy of Russians.

Period Y- is divided into two political eras: the Yeltsin era, when uncertainty grew and the socio-economic situation of the majority of the country's population worsened; and the Putin era - when certainty grew, the vertical of power was strengthened, the socio-economic situation improved, the optimism of the voting majority multiplied.

The graph clearly shows the growth of the curve since 1999, the post-default year: the pre-active demographic policy is still 8 years old.

In Russia, the process of narrowed reproduction is deepening (see Table 5.)

Table 5. Dynamics of the net reproduction rate of the population in the Russian Federation in 1960 - 2000.

The narrowed reproduction of the urban population began by the end of the 1950s, the rural population - from 1993.

In 2000, every 1000 women of fertile age left 529 girls in cities, and 704 in rural areas.

Over the reproduction of the population, it develops in parallel with the expansion of the boundaries of growth in the number of human populations as a result of the production activities of people. But the first steps on the historical path humanity makes with this type of population reproduction, which is formed "between two worlds": the goals of demographic regulation are set by nature, the means are given by society. This original ...

Over a long period of time, it allows us to conclude that as a result of changes in the course of socio-economic development and under the influence of demographic policy, there has been a transition to a new type of population reproduction with low demographic indicators, which makes it possible to weaken the pressure of the population on the productive forces and the natural environment and reduce the extent of population pressure ...

As for the frequency of birth of girls among women of different ages, then, generally speaking, it is different. However, it will not be a big mistake to assume that the proportion of girls among births is the same for all ages and is approximately 0.487-0.488. From here it is possible to obtain a summary characteristic of the fertility of the female population, which is gross factorpopulation production-the number of girls, which on average roevery woman lives during her entire reproductive period. When calculating the gross ratio, it is assumed that there is no mortality among women until the end of reproductive age.

The gross reproduction rate is equal to the total fertility rate multiplied by this proportion of girls among newborns:

where R - gross reproduction rate, TFR - total fertility rate, ASFR X - age-specific fertility rates, Δ - proportion of girls among newborns.

In our country, the average value of the proportion of girls among newborns over the past 40 years was about 0.487 (with a minimum value for these years about 0.485 and a maximum of 0.489. See also Chapter 3). If the calculation is carried out at five-year intervals, namely, data of this kind are usually available, then the formula for calculating the gross reproduction rate is as follows:

As you can see, the gross reproduction rate of the population is the total fertility rate adjusted for the secondary sex ratio.

In 1999, the value of the gross coefficient in our country was only 0.570, which means its more than two-fold decrease in the period from 1960 to 1999.

The gross reproduction rate of the population ... can be interpreted in various ways: first, as an age-standardized fertility rate ...; second, as the average number of daughters that a group of women who began life at the same time could give birth if they all lived to the end of their childbearing period; thirdly, as the ratio between the number of women of one generation, for example, at the age of 15 years, to the number of their daughters at the same age, provided that there is no mortality during the childbearing period; fourth, as the relationship between female births in two successive generations, assuming that no one dies between the beginning and the end of the reproductive period. The last three definitions are usually used when talking about real cohorts, however, any of these interpretations can be used regardless of whether the gross reproduction rate is calculated for a hypothetical generation, or for a real one. Shryock H.S., Sigel J.S. The Methods and Materials of Demography. N. Y. San Francisco, London 1973. P. 3/5.

Net reproduction rate

An indicator that also takes into account mortality is No then is the coefficient of population reproduction, or otherwise, odds Fitsi Beka-Kuchinski . Otherwise, it is called the net reproduction rate of the population. It is equal to the average number of girls born in a woman's lifetime and surviving to the end of the reproductive period, given the birth and death rates. The net reproduction rate of the population is calculated using the following approximate formula (for data for five-year age groups):

where all designations are the same as in the formula for the gross coefficient, a 5 L x f and l 0 - respectively, the number of people living in the age interval (x + 5) years from the female mortality table. The formula for calculating the net reproduction rate of the population uses the number of people living in the age interval (x + n) years from the female mortality table, and not the function of survival, that is, not the number of survivors before it begins (l x ), because it is an approximate formula. It is the survival function that is used in rigorous demographic analysis and mathematical applications of demography 1 (x).

Despite the somewhat "threatening" appearance, this formula is quite simple and allows without any special difficulties, especially using the appropriate software, for example, Excel spreadsheets, to calculate the value of the net reproduction rate of the population. In addition, many programs have been developed that allow you to reduce the calculation of the net coefficient to a simple input of initial data. For example, the International Program Center of the US Bureau of Census (IPC of the US Bureau of the Census) has developed a spreadsheet system PAS (Population Spreadsheets Analysis), one of which (SP) based on data on the values of the age-specific fertility rates and the number of people living in the age interval (x + n) years calculates gross and net reproduction rates, as well as the true rate of natural increase and generation length, which will be discussed below 3.

The calculation was made according to the following step-by-step procedure:

Step 1. In column 2 we enter the values of the age-specific fertility rates ( 5 ASFR X , taken in this case from the Demographic Yearbook of the Russian Federation for 1999 (p. 155 **).

Step 2. We calculate the total fertility rate (TFR). For this, the number in the rows of column 2 is divided by 1000 in order to express the age-specific fertility rates in relative shares of 1 (in other words, we bring these values to 1 woman of a conditional generation). We enter the obtained quotients in column 3. The sum of these numbers, multiplied by 5, gives us the value of the total fertility rate equal to 1.2415 (highlighted bold italic). This, to within the third decimal place, coincides with the official data of the State Statistics Committee of the Russian Federation (1.242. WITH. 90).

Step 3. We calculate the gross reproduction rate (TO), or the number of daughters a woman has born in her lifetime. To do this, we multiply the data in column 3 row by row by the proportion of girls among newborns In this case, its average value for the period 1960-1998 was taken, equal to 0.487172971301046. The sum of the numbers in column 4 multiplied by 5 gives the gross reproduction rate equal to 0.6048. The same result can be obtained by simply multiplying the total fertility rate by the proportion of girls among newborns (1.2415 0.487 ... = 0.6048).

Step 5. Calculate the net reproduction rate. For this, we multiply the data in column 4 row by row by the numbers in column 6. Summing up column 7, we obtain the value of the net reproduction rate equal to 0.583. This value differs only by 0.002 from that officially published by the State Statistics Committee of the Russian Federation (0.585, p. 114 of the Demographic Yearbook for 1999).

The net reproduction rate is calculated for a conditional generation. As a measure of replacement of the maternal generation by the generation of daughters, it is valid only for the so-called stable population, in which the reproduction regime does not change, i.e. fertility and mortality. The size of such a population changes (i.e. increases or decreases) in R 0 once in a while T, called the average generation length.

Calculation of indicators of reproduction of the population of Russia for 1998 5

Table 7.1

Population growth and reproduction are determined by the ratio between the number of births and deaths, or, in other words, between the levels of fertility and mortality. The word "natural", as mentioned earlier, in this case is conditional, intended to designate exactly this relationship between fertility and mortality, in contrast to changes in population due to migration processes. There are similarities and interactions between population growth and reproduction. But there is a significant difference between these concepts. In particular, the population size may continue to grow for a long time, while the reproduction of the population has already become narrowed (i.e., each subsequent generation is numerically smaller than the previous one). This situation is explained by the fact that the age structure carries some potential for demographic growth.
On the contrary, the size of the population may continue to decline under the regime of expanded reproduction (if the share of the reproductive part of the population becomes too small in comparison with the share of the elderly part of it. Then the number of births, even with a very high birth rate, would not be able to compensate for the large number of deaths). And this is explained by the same potential for population growth, which carries the age structure of the population, but already with a negative sign (in the algebraic sense).

7.1. General rate of natural growth
Population growth (or growth, which is actually the same) is characterized by a number of indicators, the simplest of which is the general coefficient of natural growth already known from Chapter 4. Let me remind you that this coefficient is the ratio of the natural increase in the population to its average (most often the average annual) number. Let me also remind you that natural increase is the difference between the number of births and deaths in the same period of time (usually in a calendar year) or the difference between the total fertility and mortality rates.
The rate of natural increase has all the same advantages and disadvantages as other general rates. Its main drawback is the dependence of the value of the coefficient and its dynamics on the characteristics of the age structure of the population and its changes. It should be noted that this dependence of the rate of natural increase on the age structure is even much more significant than other general coefficients. It doubles, as it were, by the simultaneous influence of the age structure on the levels of fertility and mortality in opposite directions. Indeed, say, in a relatively young population, with a high proportion of young people from 20 to 35 years old (when the first and second children are born, the probability of which is still quite high today, and the probability of death at these ages, on the contrary, is small) even at a moderate birth rate, there will be a relatively high number of births (due to the large number and share of young married couples in the total population) and, at the same time, for the same reason, due to the young age structure, a relatively lower number of deaths. Hence, the difference between the number of births and deaths, i.e. natural growth and the rate of natural growth. On the contrary, with a decrease in the birth rate and as a result of this reduction - an aging of the age structure - the number of deaths will increase (while the mortality rate in each age group may remain unchanged or even decrease), and ultimately the natural population growth and the rate of natural increase will decrease. ... It is the latter that is happening in our country, as well as in other economically developed countries with a low birth rate.
The dependence of the value of the general coefficient of natural growth on the age structure of the population must be taken into account in a comparative analysis when comparing such coefficients for countries or territories with populations that differ from each other in the nature of their demographic development and, accordingly, in the nature of their age structure.
One of the ways to eliminate this drawback, to bring the compared coefficients of natural increase to a comparable form, can be the index method and methods of standardization of general coefficients already known to the reader. The scope of this textbook does not allow considering these methods here (but you can get acquainted with them in reference books on statistics and in other scientific literature).
Another way to improve the quality of measuring the level of population dynamics is to move from natural growth to calculating the indicators of population reproduction. The advantage of these indicators lies in their independence from the structure of the population, primarily from gender and age.

A special method of standardizing the coefficients of natural growth is considered, in particular, in the article: Borisov V.A. Standardization of the coefficient of natural growth of the population // Demographic factors and living standards. / Ed. D.L. Broker and I.K. Belyaevsky. - M., 1973.S. 376-379.

7.2. Reproduction rates
There are several such indicators, of which two are gross and net reproduction rates. In contrast to the rate of natural increase, these indicators characterize the change in the population not over the year, but over the period of time during which the parental generation is replaced by the generation of their children. Since the replacement of generations is characterized by the ratio of fertility and mortality rates, and the latter differs significantly between males and females, population reproduction rates are calculated separately for each sex, more often for females. Usually, this does not take into account the external migration of the population, i.e. the so-called closed population (conditionally not subject to external migration) is considered.
The gross reproduction rate is calculated in the same way as the total fertility rate, but unlike the latter, only girls are taken into account in the calculation. In the form of a formula, the calculation can be presented as follows:
(7.2.1)
where r1 - gross reproduction rate of the population; TFR - total fertility rate; d is the proportion of girls among newborns.
Thus, the gross reproduction rate of the population shows the number of girls that, on average, give birth to one woman in her entire life. It is assumed that none of the women and their daughters die before the end of the reproductive period of life (conventionally - up to 50 years). Obviously, the assumption of no mortality is too unrealistic for the gross ratio to be of any use in analytical work. Indeed, in recent years this indicator has not actually been used. If we take into account the effect of mortality on the degree of population reproduction, then we turn to the net population coefficient. It is calculated using the following formula:
(7.2.2)
where R0 - Fx - FLx- the number of living women from the mortality tables, which serve as an adjustment for mortality (or for survival to a certain age, which in this case is the same); l0 - "root" of the mortality table, equal to 100,000 or 10,000, depending on its digit capacity; d is the proportion of girls among newborns; P - the length of the age range (usually either 1 or 5).
Traditionally, the coefficient is calculated on average per woman, so the formula contains a factor of 0.001. But it is possible to calculate an average of 1000 women. This, again, as in the case of the names of the indicators of population reproduction, is a matter of the user's arbitrary choice.
The net reproduction rate of the population characterizes the replacement of a generation of mothers with a generation of their daughters, but is often interpreted as an indicator of replacement of generations in the entire population (of both sexes together). If this coefficient is 1.0, this means that the ratio of birth rates and mortality rates ensures simple reproduction of the population through periods of time equal to the average age of mothers at the birth of their daughters. This average age varies slightly in direct proportion to the height of the fertility rate between 25 and 30 years. If the net coefficient is greater or less than 1.0, this means, respectively, expanded reproduction of the population (the generation of children is numerically larger than the parent) or narrowed (the generation of children, taking into account their survival to the average age of their parents, is numerically less than the parent).
The average age of mothers at the birth of their daughters (more precisely, at the birth of daughters, who, in turn, live at least to the age of their mothers at the time of their birth. But this condition is pronounced so long that almost everyone, even the most strict experts, omit it ), also called the length of the female generation, is approximately calculated by the formula:
(7.2.3)
where T - length of the female generation (average age of mothers at the birth of daughters); Fx - age-specific fertility rates; FLx - number of living women from life tables; d is the proportion of girls among newborns; X - age at the beginning of the age interval; P is the length of the age interval in years.
Since in the above formula, the indicators of the length of the age interval (P) and the proportion of girls among newborns (d) is included in both the numerator and the denominator of the fraction, they could obviously be reduced. But in practice, it turns out that you do not need to do this (the number of columns in the calculation table increases unnecessarily).
It is easy to see that the denominator of the above formula contains the expression of the net reproduction rate of the population, and in general the formula expresses the arithmetic mean of the average ages for each five-year age interval, weighted by the proportion of newborn girls who live to the age of their mothers at the time of their birth.
An example of calculating the net reproduction rate of the female population of Russia for 1996 and the average age of mothers at the birth of their daughters is shown in Table 7.1.
Consider the calculation algorithm by its stages:
1) are written out from the Demographic Yearbook of Russia (M., 1997, p. 215) in column 1 of table 7.1 age-specific fertility rates, while they are converted from ppm to unit fractions (by dividing each by 1000);
2) multiplying each of the age-specific fertility rates by the proportion of girls among newborns (assuming it to be the same in all age groups of mothers), we obtain the age-specific fertility rates for girls, which are recorded in column 2;
3) according to the tables of mortality of the population of Russia for 1996 (See Demographic Yearbook of Russia. M., 1997, p. 250), the number of people living in each age group is determined as the arithmetic mean of two adjacent numbers of survivors, i.e.:

where FLx- the number of women living, calculated according to mortality tables; lx and lx + 5- the number of survivors X and x + 5 from the same mortality tables.
The numbers of living obtained in this way are divided by the root of the mortality table l 0 (in this case it is equal to 100,000) and are entered in column 3 of Table 7.1;
5) the age-specific fertility rates of girls from column 2 are multiplied line by line by the number of living women from column 3 (i.e., thus, an amendment is made for their survival to the age of the mothers at which they gave birth to these daughters). The multiplication results are recorded in column 4;
6) the indicators of columns 1, 2, and 4 are summed vertically, and the sums are multiplied by 5 (by the length of the age intervals). As a result, in column 1, the total fertility rate is obtained TFR = 1.2805, or rounded 1.281; in column 2, the gross reproduction rate of the population, equal to 0.625, and in column 4 - the net reproduction rate of the population R0 = 0.60535, or rounded 0.605.
Naturally, it is interesting to compare the results obtained with the official publications of the Goskomstat of Russia, which are calculated in the most accurate way based on one-year age coefficients. It turned out that the total fertility rate calculated by us in Russia for 1996 exactly coincided in magnitude with the calculated by the State Statistics Committee of Russia - 1.281. The value of the net coefficient differed from the calculations of the State Statistics Committee by only 0.002. This discrepancy can be considered insignificant.
Let's return to table 7.1 and now we will determine the average age of mothers at the birth of their daughters - the length of the female generation. For this you need:
7) multiply row by row the data in column 4 by the age indicators in the middle of each five-year age interval (in column 5), and write the results of this multiplication in column 6. After summing the products obtained and multiplying the sum by 5, we obtain the numerator of the fraction (15.1237), dividing which by the net reproduction rate of the population (0.60535), we obtain the indicator of the length of the female generation in Russia in 1996, equal to 24.98 years (or, rounded off, 25 years).
The net reproduction rate of the population makes it possible to assess the state of the population reproduction regime actually existing at any given moment of time (the ratio of fertility and mortality rates in their distraction from the impact of the age and sex structure of the population) from the standpoint of its probable further development. It characterizes not the current demographic situation, but its limiting state in some future, if the given mode of reproduction remains unchanged. In other words, the net ratio is a tool for assessing the situation and forecasting its future trends.

Table 7.1

Calculation of the net reproduction rate of the population

Russia for 1996 and the average age of mothers at
the birth of daughters

Age groups (years)	Fx/ 1000	Gr. 1 x x 0.488	(column 2 x column 3)	x + 0.5n	(x + 0.5p) X

Based on the net ratio and the length of the female generation, it is possible to determine the so-called true rate of natural population growth, which characterizes the growth of the population for each year, but, like the net coefficient, does not depend on the peculiarities of the age structure of the population. The true rate of natural population growth is approximately determined by the formula proposed by the American demographer Ansley Cole in 1955:
(7.2.4)
where r - true rate of natural population growth; R0 - net reproduction rate of the population; T - length of the female generation (average age of mothers at the birth of daughters).
Let us define, for example, this coefficient for Russia in 1996 according to the data in Table 7.1.
- (minus) 20.1 ‰.
The actual coefficient of natural growth of the population of Russia in 1996 was -5.3 ‰. From this we can see what role its age structure continues to play in the growth of our population and what the annual decline of our population will be when the age structure finally loses its potential for demographic growth.
In 1996, an interesting and simple method for assessing population reproduction was proposed by the Russian demographer V.N. Arkhangelsk. The method consists in determining the hypothetical birth rate required to ensure zero natural population growth in conditions of the actual mortality rate and the real age structure of the population. The hypothetical birth rate in this case is expressed by the total fertility rate.
The proposed method is easier to show with a specific example. As you know, the natural increase is equal to zero in the case of equality of the number of births and deaths (and, accordingly, the total fertility and mortality rates). In 1996, the crude death rate in Russia was 14.2. Therefore, to ensure zero growth, the total fertility rate would have to be the same, i.e. 14.2. In fact, its value in the same 1996 was only 8.9, or 1.6 times less. Since the age structure in this case is taken as it really is, it turns out that in order for the total fertility rate to equal the general mortality rate, it is necessary to increase the age-specific fertility rates and, as a result, the total fertility rate also by 1.6 times. compared to the actual one.
The actual total fertility rate in Russia in 1996 was 1.281 children (per woman). From here we can determine the value of the total fertility rate, which, given the current mortality rate and the current age structure of the population, could provide zero population growth in our country. This value should be 2.05 for 1996 conditions. Not a very large value, which indicates a positive (for 1996 conditions) influence of the age structure of the population. By the way, this positive influence of the age structure also points to the right time for activating the pro-natalist (i.e., aimed at stimulating the birth rate) demographic policy. The effect could be achieved at a lower cost.
Although the described method of V.N. Arkhangelsky is very simple, he quite well reveals the scale of the task facing our entire society to overcome the demographic crisis.

Some experts prefer to call these indicators “gross” and “net” reproduction rates (instead of “gross” and “net”, respectively). It seems to me that there are no serious reasons in favor of the preference for the names of reproduction indicators. I think this is just a matter of personal taste. The names I have chosen seem preferable, only because they have fewer associations with other familiar concepts.

See Family and family policy in the Pskov region / Ed. N.V. Vasilyeva and V.N. Arkhangelsk. - Pskov, 1994.S. 180-181.

7.3. Fertility Ratio
and mortality in the dynamics of population reproduction
The issue of the role of fertility and mortality in the reproduction of the country's population in recent years is being discussed among domestic specialists. Which problem is more acute: low fertility or relatively high mortality? What problem should be addressed first? Meanwhile, the answer to this question is not difficult, as it seems to me, to obtain with the help of the index method already known to us. Let's return again to the net reproduction rate of the population. It is the best indicator of population reproduction precisely because it develops as the ratio of only two components of fertility and mortality. Other factors, primarily the age structure of the population, are not present in the formula for calculating it. Hence, using a simple system of indices, it is possible to show to what extent the change in the value of the net coefficient for any period of time is due to changes in the birth rate, and to what extent - to mortality.
Let us consider the change in the net reproduction rate of the population of Russia for the period from 1986-1987. through 1996 inclusive. The choice of this period is due to the following circumstances. Increasing since the end of the 1970s, the net ratio reached by 1986-1987. maximum (1.038), and then began to decline, reaching in 1996 the value of 0.603.
Let us construct a system of indices characterizing the components of the change in the net reproduction rate of the population of Russia for the period from 1986-1987 to 1996, using its standard formula (7.2.2).

(7.3.1)
For the calculation, it turns out to be sufficient to calculate only one element of equation (7.3.1), which is the net coefficient at the level of age-related fertility in 1996 and mortality in 1986-1987. (i.e., assuming that the mortality rate remains unchanged in the 1986-1996 decade).
Turning again to the system of indices (in the rightmost part of equation 7.3.1), we note that the first of the two indices characterizes the change in the value of the net coefficient due to changes in fertility, the second - due to changes in mortality.
The calculation results are presented in table 7.2. With our accepted hypothesis about a constant mortality rate in 1986-1987. and the actual birth rate in 1996, the net reproduction rate of the population would have been 0.606 in 1996. In fact (that is, with the actual death rate in 1996), it was equal to 0.603. Already from this, frankly, insignificant difference, one can draw a conclusion about the role of an increase in mortality in the decade we are analyzing. But let's bring our calculation to the end.

Table 7.2

Calculations of the net reproduction rate

the population of Russia at the birth rate of 1996 and
various hypotheses about the mortality rate

Age group (years)	Age fertility rates in 1996 Fx *1996* / 1000	Five-year sums of the numbers of living women from mortality tables with different average life expectancy at birth		F X x FL X
Age group (years)		74.6 years (1986-1987)	80.0 years (typical tables)	gr... I xGp. 2	gr... I xGp. 3









	R0 =

Let us substitute the known and calculated values of the net coefficients into the system of indices (7.3.1):

Subtracting the obtained indices from 1, and converting the results into percentages, we determine the change in the net coefficient in structural terms:
-41,9% = -41,6% - 0,5%.
After adjusting, we get: -41.9% = - 41.4% - 0.5%.
The final conclusion: for the period under review 1986-1996. the net reproduction rate of the population of Russia decreased by 41.9% as a whole, including by 41.4% due to a decrease in the birth rate and by 0.5% due to an increase in mortality. If we take the overall decrease in the net coefficient as 100%, then 98.8% of this decrease is due to a fall in the birth rate and only 1.2% - to an increase in mortality.
Now suppose that the average life expectancy of Russian women would suddenly rise to what has already been achieved in a number of advanced countries in this respect - up to 80 years (this is the level reached in Scandinavia, France, surpassed in Japan), but the birth rate would remain at the level of 1996. Then the value of the net coefficient would be 0.621 (column 5 of Table 7.2.), i.e. would increase by only 3.0% compared to the actual value in 1996.
From this simple calculation, one can see that the role of today's, not very favorable, mortality in our country in changes in population reproduction is very small. By this I do not want to belittle the significance of the struggle against death. No, of course, social, economic, political, etc. The significance of this struggle is undeniable. But the demographic value turns out to be negligible. Today, the main factor on which the demographic future of our country depends entirely is the birth rate.