Population growth and reproduction are determined by the ratio between the numbers of births and deaths, or, in other words, between the birth and death rates. The word “natural,” as mentioned earlier, in this case is of a conditional nature, intended to designate precisely 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 may continue to grow for a long time, while population reproduction 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 with it some potential for demographic growth.
On the contrary, the population may continue to decline even under a regime of expanded reproduction (if the share of the reproductive part of the population becomes too small compared to the share of the elderly part. Then the number of births, even at 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 is carried by the age structure of the population, but with a negative sign (in the algebraic sense).

7.1. General rate of natural increase
Population growth (or growth, which is actually the same thing) is characterized by a number of indicators, the simplest of which is the general coefficient of natural increase, already known from Chapter 4. Let me remind you that this coefficient is the ratio of the magnitude of natural population growth to its average (most often 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 a calendar year) or the difference between the crude birth and death rates.
The natural increase rate has all the same advantages and disadvantages as other general rates. Its main drawback is the dependence of the coefficient value 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 coefficient of natural increase on the age structure is even much more significant than other general coefficients. It is, as it were, doubled by the simultaneous influence of the age structure on the levels of fertility and mortality in opposite directions. In fact, say, in a relatively young population, with a high proportion of young people from 20 to 35 years old (when first and second children are born, the probability of birth of which is still quite high today, and the probability of death at these ages, on the contrary, is small), even with a moderate level of fertility, a relatively high number of births will be observed (due to the large number and proportion 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 smaller number of deaths. Hence, the difference between the number of births and deaths will be correspondingly greater, i.e. natural increase and natural increase rate. On the contrary, with a reduction in the birth rate and as a result of this reduction - an aging age structure - the number of deaths will increase (while the mortality rate in each age group may remain unchanged or even decrease), and ultimately 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 low birth rates.
The dependence of the value of the general coefficient of natural increase 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 shortcoming and bring the compared natural increase coefficients to a comparable form is the index method and methods for standardizing general coefficients already known to the reader. The scope of this textbook does not allow us to consider these methods here (but they can be found 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 increase to calculating indicators of population reproduction. The advantage of these indicators is their independence from the structure of the population, primarily from gender and age.

The method of standardization of natural growth rates is specifically discussed, in particular, in the article: Borisov V.A. Standardization of the natural population growth rate // Demographic factors and living standards. /Ed. D.L. Broker and I.K. Belyaevsky. - M., 1973. S. 376-379.

7.2. Population reproduction indicators
There are several such indicators, two of them are gross and net population reproduction rates. Unlike the rate of natural increase, these indicators characterize the change in population not over a year, but over a period of time during which the parent generation is replaced by the generation of their children. Since generation replacement is characterized by the ratio of fertility and mortality levels, and the latter differs significantly between males and females, population reproduction rates are calculated separately for each sex, more often for females. Usually, external migration of the population is not taken into account, i.e. the so-called closed population (conditionally not subject to external migration) is considered.
The gross population 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 represented as follows:
(7.2.1)
Where r1 - gross population reproduction rate; TFR - total fertility rate; d is the proportion of girls among newborns.
Thus, the gross population reproduction rate shows the number of girls that an average woman gives birth to in her entire life. It is assumed that none of the women and their daughters die until the end of the reproductive period of life (conditionally - up to 50 years). Obviously, the assumption of no mortality is too unrealistic for the gross rate to be of any usefulness for use in analytical work. Indeed, in recent years this indicator has not actually been used. If we take into account the influence of mortality on the degree of population reproduction, then we move on 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 mortality tables, which serve as an adjustment for mortality (or survival to a certain age, which in this case is the same thing); l0 - the “root” of the mortality table, equal to 100,000 or 10,000, depending on its digit; d is the proportion of girls among newborns; P - length of the age interval (usually either 1 or 5).
Traditionally, the coefficient is calculated on average per woman, so the formula contains a multiplier of 0.001. But it is possible to calculate on average per 1000 women. This, again, as in the case of the names of population reproduction indicators, is a matter of arbitrary choice by the user.
The net replacement rate of the population characterizes the replacement of the generation of mothers with the generation of their daughters, but is often interpreted as an indicator of the replacement of generations in the entire population (both sexes together). If this coefficient is equal to 1.0, this means that the ratio of fertility and mortality levels ensures simple reproduction of the population over periods of time equal to the average age of mothers at the birth of daughters. This average age varies slightly in direct proportion to the height of the birth rate, ranging between 25 and 30 years. If the net coefficient is more or less than 1.0, this means, respectively, expanded population reproduction (the generation of children is numerically larger than the parent’s) or narrowed (the generation of children, taking into account their survival to the average age of their parents, is numerically smaller than the parent’s).
The average age of mothers at the birth of 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 so long to pronounce that almost everyone, even the most strict experts, omit it ), also called the length of the female generation, 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 mortality tables; d is the proportion of girls among newborns; X - age at the beginning of the age interval; P- 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 denominator of the fraction; they could obviously be reduced. But in practice it turns out that this is not necessary (the number of columns in the calculation table increases unnecessarily).
It is easy to notice 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 surviving 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 daughters is given in Table 7.1.
Let's consider the calculation algorithm in its stages:
1) age-specific birth rates are written out from the Demographic Yearbook of Russia (M., 1997, p. 215) in column 1 of Table 7.1, and they are converted from ppm to fractions of a unit (by dividing each by 1000);
2) multiplying each of the age-specific birth rates by the share of girls among newborns (assuming it to be the same in all age groups of mothers), we obtain age-specific birth rates for girls, which are recorded in column 2;
3) according to the mortality tables of the population of Russia for 1996 (See Demographic Yearbook of Russia. M., 1997. P. 250), the numbers of people living in each age group are determined as the arithmetic mean of two adjacent numbers of those living, i.e.:

Where FLx- the number of living women, calculated from mortality tables; lx And lx+5- number of people living to age X And x+5 from the same mortality tables.
The numbers of living people obtained in this way are divided by the root of the mortality table l 0 (in this case it is equal to 100000) and are entered in column 3 of table 7.1;
5) age-specific birth rates for girls from column 2 are multiplied line by line by the number of living women from column 3 (i.e., in this way an adjustment 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) indicators in columns 1, 2, and 4 are summed vertically, and the sums are multiplied by 5 (by the length of age intervals). As a result, the total birth rate is obtained in column 1 TFR = 1.2805, or rounded 1.281; in column 2 the gross population reproduction rate is equal to 0.625, and in column 4 - the net population reproduction rate R0 = 0.60535, or rounded to 0.605.
Naturally, it is interesting to compare the results obtained with official publications of the State Statistics Committee of Russia, which are calculated in the most accurate manner based on one-year age coefficients. It turned out that the total fertility rate we calculated for Russia for 1996 exactly coincided in value with that calculated by the State Statistics Committee of Russia - 1.281. The value of the net coefficient differed from Goskomstat calculations by only 0.002. This discrepancy can be considered insignificant.
Let's return to Table 7.1 and now determine the average age of mothers at the birth of daughters - the length of the female generation. To do this you need:
7) multiply the data in column 4 line by line 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 resulting products and multiplying the sum by 5, we obtain the numerator of the fraction (15.1237), dividing which by the net population reproduction rate (0.60535), we obtain an indicator of the length of the female generation in Russia in 1996 equal to 24.98 years (or rounded - 25 years).
The net population reproduction rate makes it possible to assess the state of the population reproduction regime actually existing at any given moment in time (the ratio of birth and death rates in their abstraction from the impact of the age-sex structure of the population) from the standpoint of its probable further development. It characterizes not the current demographic situation, but its ultimate state in some future if the given reproduction regime remains unchanged. In other words, the net coefficient is a tool for assessing the situation and forecasting its future trends.

Table 7.1

Calculation of the net population reproduction rate

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

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

Based on the net coefficient and the length of the female generation, the so-called true rate of natural population growth, which characterizes population growth for each year, but, like the net coefficient, does not depend on the characteristics of the age structure of the population. The true rate of natural population growth is approximately determined by the formula proposed by American demographer Ansley Cole in 1955:
(7.2.4)
Where r - true rate of natural population growth; R0 - net population reproduction rate; T - length of the female generation (average age of mothers at the birth of daughters).
As an example, let us determine this coefficient for Russia in 1996 according to Table 7.1.
-(minus) 20.1 ‰.
The actual rate of natural population growth in Russia in 1996 was -5.3‰. From this we can see what role our 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 Russian demographer V.N. Arkhangelsk. The method consists in determining the hypothetical birth rate required to ensure zero natural population growth in the context of the actual mortality rate and the actual 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 demonstrate with a specific example. As is known, natural growth is zero if the numbers of births and deaths are equal (and, accordingly, the overall birth and death rates). In 1996, the overall mortality rate in Russia was 14.2. Consequently, 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 accepted as it actually is, it turns out that in order for the total fertility rate to be equal to the total mortality rate, it is necessary to increase the age-specific birth rates and, as a result, the total fertility rate also by 1.6 times compared to actual.
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 ensure 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 indicates the right time to intensify pronatalist (i.e., aimed at stimulating the birth rate) demographic policy. The effect could be achieved at lower cost.
Although the described method by V.N. Arkhangelsky is very simple; it quite well reveals the scale of the task that faces our entire society in overcoming the demographic crisis.

Some experts prefer to call these indicators “gross” and “net” population reproduction rates (instead of “gross” and “net”, respectively). It seems to me that there are no serious grounds for preferring the names of reproduction indicators. I think it's 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. Arkhangelsky. - Pskov, 1994. P. 180-181.

7.3. Birth rate ratio
and mortality in the dynamics of population reproduction
Among domestic experts, the issue of the role of fertility and mortality in the reproduction of the country's population in recent years is being discussed. Which problem is more acute: low fertility or relatively high mortality? What problem should be solved first? Meanwhile, it seems to me that the answer to this question is not difficult to obtain using the index method already known to us. Let's return again to the net population reproduction rate. It is the best indicator of population reproduction precisely because it develops as a 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. From here, using a simple system of indices, it is possible to show to what extent the change in the value of the net coefficient over any period of time is due to changes in the birth rate, and to what extent - to the mortality rate.
Let us consider the change in the net reproduction rate of the Russian population over the period from 1986-1987. up to 1996 inclusive. The choice of this period is due to the following circumstances. Increasing since the late 1970s, the net ratio reached by 1986-1987. maximum (1.038), and then began to decline, reaching a value of 0.603 in 1996.
Let us construct a system of indices characterizing the components of changes 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 is sufficient to calculate only one element of equation (7.3.1), which is the net coefficient at the level of age-specific fertility in 1996 and mortality in 1986-1987. (i.e., assuming a constant mortality rate in the decade 1986-1996).
Turning again to the system of indices (on the right extreme side 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 the birth rate, the second - due to changes in mortality.
The calculation results are presented in Table 7.2. Under our accepted hypothesis of a constant mortality rate in 1986-1987. and the actual birth rate in 1996, the net population reproduction rate would have been 0.606 in 1996. In fact (i.e., with actual mortality in 1996) it was equal to 0.603. Already from this, frankly, insignificant difference, we can draw a conclusion about the role of the increase in mortality in the decade we are analyzing. But let's bring our calculation to the end.

Table 7.2

Net reproduction rate calculations

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

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









	R0 =

Let's substitute the known and calculated values of net coefficients into the index system (7.3.1):

Subtracting the resulting 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 adjustment we get: -41.9% = - 41.4% - 0.5%.
Final conclusion: for the period under review 1986-1996. The net reproduction rate of the Russian population decreased by a total of 41.9%, 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% is due 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 regard - up to 80 years (this is the level achieved in Scandinavian countries, in France, surpassed in Japan), but the birth rate would remain at the 1996 level. Then the value of the net coefficient would be 0.621 (column 5 of table 7.2.), i.e. would have increased by only 3.0% compared to the actual figure in 1996.
From this simple calculation we can see that the role of today’s not very favorable mortality rate in our country in changes in population reproduction is very small. By this I do not at all want to diminish the importance of the fight against death. No, of course, social, economic, political, etc. The significance of this struggle is undeniable. But the demographic significance turns out to be negligible. Today, the main factor on which the demographic future of our country entirely depends is the birth rate.

Concept of population reproduction

Topic 11. Population reproduction

The main feature of the population is that, despite constant changes in its size and structure, it remains as a population, i.e. as a self-reproducing collection of people . One can even say that the population self-preserves, remains itself precisely and exclusively thanks to these continuous changes.

This process of self-preservation of the population in the course of its continuous changes is called population reproduction, and it is this process that forms the subject of demography as a science.

Population reproduction- this is a constant renewal of the number and structure of the population in the process of changing generations of people, through births and deaths. The set of parameters that determine this process is called population reproduction regime.

The parameters that determine population reproduction are fertility and mortality, presented in the form of their own indicators, as well as the number of arrivals and the number of departures1.

Typically, population reproduction is considered not as a whole, but in relation to any one sex, most often female. The choice of the female population is due to the following factors:

· the reproductive period of women is shorter than that of men;

· the basic parameters of female reproduction (the number of children born to a woman, her age at their birth, etc.) are much more accessible than similar characteristics for men, especially with regard to out-of-wedlock births.

The role of age as a universal independent variable in demographic analysis and its constant change (every person inevitably either dies or becomes older, i.e., more strictly speaking, moves to another age group) determine that in the analysis of population reproduction much attention is paid to age , studying this process across age groups.

Population reproduction indicators refer to a real or hypothetical cohort (generation), i.e. are essentially cohort.

If certain gender- and age-differentiated fertility and mortality rates are given, as well as a secondary sex ratio, which is a universal biological constant and is equal to approximately 105-106 live births of boys per 100 live births of girls, then this completely determines the reproduction of the population and its age-sex structure. It is the totality of these parameters that is meant when speaking about the population reproduction regime.

Since the reproduction of the female population is usually studied, the whole question comes down to considering the age-specific mortality of women and the frequency of births of girls among women of different ages.

Mortality is typically measured using the survival-to-age function X years, i.e. using the function . In practice, they use the numbers of people surviving to age X years from complete mortality tables of the female population. A general characteristic of female mortality is the average life expectancy of a newborn, i.e. .

Gross population reproduction rate- this is the number of girls that on average each woman will give birth to during her entire reproductive period. When calculating the gross coefficient, it is assumed that there is no mortality among women until the end of their reproductive years.

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

Where R- gross reproduction rate; TVR - total fertility rate; ASVR x - age-specific fertility rates; - the proportion of girls among newborns.

In Russia, the average value of the proportion of girls among newborns over the past 40 years was approximately 0.487.

As can be seen from the calculation formula, the gross population reproduction rate is the total fertility rate adjusted for the secondary sex ratio.

The gross population reproduction rate can be interpreted in different ways:

· as an age-standardized birth rate;

· as the average number of daughters that a group of women who began life at the same time could give birth to if they all lived to the end of their childbearing period;

· 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 within the childbearing period;

· as the ratio between female births in two successive generations, assuming that no one dies between the beginning and end of the reproductive period.

The last three definitions are usually used when talking about real cohorts.

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 in R times more or less than the size of the mothers' generation. After all, not all of these daughters will live to reach the age their mothers were at the time of birth. And not all daughters will survive to the end of their reproductive period. This is especially true for countries with high mortality, where up to half of newborn girls may not survive to the beginning of the reproductive period, as was the case, for example, in Russia before the First World War 2 . Nowadays, of course, this no longer exists (in 1997, almost 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. Given the assumption of zero mortality until the end of the reproductive period, the gross population reproduction rate has recently been practically not published or used.

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

where all the notations 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 at the age interval (x+n) years from the female mortality table, and not a function of survival, i.e., not the number of people surviving until it begins (l x), because this is an approximate formula. In rigorous demostatistical analysis and mathematical applications of demography, it is the survival function that is used 1(x).

Despite its somewhat “threatening” appearance, this formula is quite simple and allows you to calculate the net reproduction rate without much difficulty, especially using appropriate software, such as Excel spreadsheets. In addition, many programs have been developed that allow you to reduce the calculation of the net coefficient to simply entering the initial data. For example, the International Program Center of the U.S. Bureau of the Census (IPC of U.S. Bureau of the Census) has developed a system of electronic tables PAS (Population Spreadsheets Analysis), 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.

In table 7.1 shows an example of calculating the age-specific birth rate, gross and net population reproduction rates, in which the above software is not used. Using this example, as well as a similar example given in the textbook by V.A. Borisov 4, you can easily learn to calculate all the main indicators of population reproduction. But, of course, it is advisable to have at least some computer equipment; it is best, of course, to use Excel.

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

Step 1. In column 2 we enter the values of age-specific birth 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 number in the lines of column 2, we divide by 1000 in order to express age-specific fertility rates in relative fractions of 1 (in other words, we reduce these values to 1 woman of a conditional generation). We enter the resulting 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, up to 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 born to a woman during her lifetime. To do this, we multiply the data in column 3 line by line by the share 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 of 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 to average 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 resulting averages by 5, we determine the number of people living at each age interval necessary for the calculation.

Step 5. We calculate the net reproduction rate. To do this, we multiply the data in column 4 line by line by the numbers in column 6. Summing up column 7, we obtain a 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 the 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. birth rate and death rate. The size of such a population changes (i.e. increases or decreases) in R0 once in a while T, called average generation length.

Calculation of indicators of population reproduction in 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 mothers at the birth of daughters who live at least to the age their mothers were at the time of their birth.

To calculate generation length, you can use an approximate formula, which is given in many demography textbooks 6:

where all the notations are the same as in the previous formula. As can be seen from the formula, the required 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.), weighted by the number (proportion) of the latter surviving at least to the age at which their mothers were at the moment of their birth. Please note that calculating generation length is completely similar to calculating the average age at birth of a child, which we did in the chapter on fertility. The only difference is in the scales used (when calculating the average age at birth of a child, as you remember, age-specific birth 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 survive at least to the age of their mother at their birth.

Let us now return again to the table. 7.1 and take the last, sixth step.

Step 6. We calculate generation length, or the average age of a mother at the birth of daughters who live at least to the age their mothers were at the time of their birth. To do this, multiply the numbers in the lines of column 7 by the middle of each age interval (column 8) and enter them in column 9. The resulting products represent the number of man-years lived by all daughters born to 1 woman of a conventional generation in a given age interval and surviving at least to the age of their mother at the time of their birth. Summing these products, we obtain the numerator of the above formula for calculating generation length, approximately equal to 14.8709. This number is the number of person-years lived by all daughters born to 1 woman of a conventional generation throughout her life and surviving at least to the age of the mother at the time of their birth. Dividing this last value by the number of all such daughters, i.e. 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 years, or rounded 25 ,38 years old.

True rate of natural increase As mentioned above, the net population reproduction rate (R0) shows that the size of a stable population corresponding to the real one with given general fertility and mortality rates, which are assumed unchanged, changes (i.e. increases or decreases) in R 0 times per time T, i.e., for the length of the generation. Taking this into account and accepting the hypothesis of exponential population growth (decrease), we can obtain the following relationship connecting the net coefficient and generation length. This relationship is derived from the following equation: Р Т = Р () R 0 = Р 0 - e g T (remember Chapter 3, the section that talks about growth and population growth rates):

In the theory of stable population, r in these expressions is called the true coefficient of natural population growth (or the A. Lotka coefficient). This coefficient represents the root of the so-called integral equation of population reproduction, or the Lotka equation 7. It is widely used in mathematical applications of demography, in particular in the theory of stable populations. However, we do not consider this equation here, since this topic is beyond the scope of our manual. Those interested are referred to the Demography Course, ed. AND I. Boyarsky (M, 1985, pp. 90-91 and 103-118), as well as to the corresponding articles of the Demographic Encyclopedic Dictionary (M., 1985) and the Encyclopedic Dictionary “Population” (M, 1994). For a very close approximate solution of the Lotka equation regarding 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)... In 1907 he showed that a population growing at a constant rate and maintaining a constant order of extinction tends to a certain age composition and is constant/ and fertility and mortality rates. ...For the first time he proposed a mathematical expression for the own coefficient of natural increase of a closed population with a constant order of extinction and childbirth, the algebraic expression of which was given in the work “On the true coefficient of natural increase of the population” (1925), showing the connection of this coefficient with the net reproduction rate of the population. .. Lotka studied the process of generational change, gave a modern analytical expression for the length of a generation...

Population. Encyclopedic Dictionary. M., 1994. P. 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 coefficients” 8, can be used to estimate the true coefficient of natural population growth, taking into account that, as stated above, the length of a generation is the average the age of the mother at the birth of daughters who survive at least to the age their mothers were at the time of their birth. In modern conditions, the length of a generation does not differ too noticeably from the average age of a mother at the birth of a child*. Therefore, estimating the last parameter in any way makes it possible to approximately determine both the sign and the magnitude of the true coefficient of natural increase.

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), we will obtain the true rate of natural population growth in Russia for 1998 conditions. This value is equal to -0.0210636435922121, or = -2.1%.

The real value of the coefficient of natural population growth in 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 Russian population, which, in turn, is associated with a slight increase in the birth rate in the first half of the 80s. last century and with the influence of previous demographic waves. The real age structure of our country is younger than the age structure of a stable population corresponding to modern parameters of fertility and mortality. The population has accumulated some growth potential, or, more precisely, the potential to slow down population decline, due to which the population of our country is not declining as quickly as it would otherwise be the case.

But this situation will end very soon. Generations born during the period of fertility decline that began in the second half of the 80s will begin to enter reproductive age. last century and continues 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 hope will not save our country from the horrors of depopulation.

For example, in the same 1998, the average age of a mother at the birth of a child, according to S.V. Zakharov, was 25.34 years. See: Population of Russia 1999. Seventh annual demographic report / Rep. ed. A.G. Vishnevsky. M., 2000. P. 55. The State Statistics Committee 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 last two years is nothing more than an artifact.

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

The clarification “after a time equal to the length of a generation” is very significant. If R0< 1, this does not mean that in the year for which the net reproduction rate is calculated, there is a reduction in population, absolute numbers of births and total fertility rate. The population can grow for quite a long time, despite the fact that the net coefficient is less than or equal to 1. This has been the case, for example, in Russia since the late 60s. until 1992. The value of the net coefficient in our country all these years was less than 1; accordingly, 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 precisely in 1992), the birth rate became less than the death rate, and the population began to decline in numbers.

We can say that depopulation in Russia has gone from hidden and latent to obvious and open. And this was completely independent of the specific political and socio-economic situation of the 90s. last century, no matter what the so-called “nationally concerned scientists” and self-proclaimed “patriots” of any color, from the ultra-left to the ultra-right, say. The beginning of depopulation in our country was predetermined by the processes that occurred in the population throughout the 20th 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, happens in all developed countries. About a third of the world's countries have a birth rate that is less than what is necessary for simple population reproduction. In other words, in these countries, as in Russia, there is a hidden or obvious 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 level of birth rate necessary to ensure simple reproduction of the population. In this regard, the question arises of how to determine this level of fertility. To answer it, different methods are used.

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

Where TFR h, TFR a, GMR, GBR- respectively, the hypothetical total birth rate necessary to ensure simple reproduction, the current total birth rate, the total mortality rate and the total birth rate.

Gross and net coefficients make it possible to do otherwise, but it is also quite simple to answer this question. To do this, use either the ratio of the net coefficient to the gross coefficient, or the inverse ratio.

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

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

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

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

Calculation using the method of V.N. Arkhangelsky gives the value of the total fertility rate necessary to ensure simple reproduction, approximately equal to 2.04, which is significantly 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 also takes into account the role of the age structure. 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 using the method of V.N. Arkhangelsky, in 1996 was approximately 2.05, i.e. we have a decrease of 0.01 over two years. Calculation using an alternative method gives for 1996 the value TFR h, equal to 2.12, which, on the contrary, is 0.01 more than 11. As we can see, the dynamics of the hypothetical total fertility rate, calculated by various methods, turned out to be the opposite. Given the declining mortality rate during that period, this difference can be explained both by some 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 such 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 accordingly measured by a gross coefficient, and “results” by a net coefficient. Moreover, the lower the p value and the closer it is to 1, the more “economical” the population reproduction is 12 . The application of supposedly “economic” terminology to population reproduction seems somewhat strange (it is not clear what to do with ethics). In addition, it seems that the name of this indicator (“price of simple reproduction”), and its interpretations in the mouths of many of our demographers are needed only to prove to ourselves and our readers that the situation with reproduction in our country is far from one that could cause alarm. What, exactly, to worry about if the value of p in our country is almost the same as in advanced Western countries. We, so to speak, if not ahead of the rest of the planet then, at least in the forefront progressive humanity.

To be involved in progress is, of course, impressive. But the question arises: is this progress? Can an inexorable and rapid fall into the abyss of depopulation be called progress? Unfortunately, many demographers either ignore these damned questions, or are at best conciliatory about the negative demographic dynamics in our country, and at worst, even considering current demographic trends (especially the situation with the birth rate) as something completely normal.

All population reproduction indicators described above refer to the female population. However, in principle, similar indicators (gross and net reproduction rates, true rate of natural increase, male generation length, etc.) can be calculated for the male population, as well as for the entire population. Analysis of the reproduction of the male population has become increasingly widespread in demography in recent years. We have already discussed above 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

Population reproduction, replacement of generations, reproduction mode, vitality index, gross coefficient, net coefficient, stable population, true rate of natural increase, Lotka coefficient, generation length, simple reproduction, narrowed reproduction, expanded reproduction, price of simple reproduction.

Review questions

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

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

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

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

What the net population reproduction rate says and doesn’t say

Apart from the completely illiterate, those who talk about the demographic situation on the basis of general birth and death rates, then most people who are more or less seriously interested in demography know that in order to correctly judge what is happening, it is necessary to use more subtle measures . These include, in particular, the total fertility rate, life expectancy and other functions of mortality tables, as well as gross and net reproduction rates.

Analysis of these indicators and their dynamics allows us to judge the changing reproductive situation, comprehend the various components of this situation and makes it possible to compare the conditions of population reproduction of countries or regions in time and space.
At the center of such an analysis is an indicator well known to demographers - the net coefficient (net coefficient) of reproduction of the female population. It is equal to the number of girls born in a given period (usually one year, but another period can be chosen, for example, a five-year period, as is done in Table 1) and who have a chance of surviving - at the age-specific mortality rates of this period - to the average age of motherhood, calculated for the same period, per woman. The components of calculating the net coefficient for five-year periods, starting from the last five-year period of the 19th century and ending with the last five-year period of the 20th century, are given in Table. 1, changes in the net coefficient itself are also shown in Fig. 1. The red line in the figure is the line of simple reproduction, the boundary separating expanded reproduction from narrowed reproduction.

The last column of the table indicates the so-called “true” coefficient of natural increase, i.e. the rate of natural increase of a stable population corresponding to the age-specific functions of fertility and mortality of each period. It shows with what annual coefficients the population can increase (decrease) due to natural growth if a constant regime of fertility and mortality for the calculation period indicated in the first column of the table is maintained indefinitely.

Table 1. Components of the net reproduction rate of the female population and the “true” rate of natural increase in Russia over 100 years

Period	Average number of children per woman	Including girls	Average age of mother, years	Probability of surviving to maternal middle age*	Net reproduction rate (2x4)	True coefficient of natural growth, ‰

At the end of the 19th century - in the first decade of the 20th century, at best, only half of girls born reached the average age of motherhood, however, with a birth rate of 7 or more children per woman, expanded population reproduction was steadily ensured in Russia - each new generation of girls was approximately 1.5 times larger than the maternal generation (the net reproduction rate fluctuated in the range of 1.5-1.6). As a result, the population could increase annually by 1.4 - 1.6% (the true rate of natural increase was 14.0 -15.5 ppm). The slow decline in fertility at that time was compensated by a gradual improvement in the survival of child generations, so that the integral indicators of reproduction changed little.

Figure 1. Net reproduction rate of the Russian population throughout the twentieth century

The smooth change in indicators is interrupted by the First World War and the Civil War and the accompanying famines and epidemics. The fall in the birth rate and the sharp deterioration in the mortality situation caused a short-term demographic crisis. If the reproduction regime indicators recorded in 1915-1919 were maintained for a long time, the population of Russia would decline by 0.4% per year. A compensatory increase in the birth rate and noticeable successes in reducing mortality in the 1920s again restored the previous characteristics of population reproduction. The value of the net reproduction rate, calculated for 1925-1929, turns out to be even higher than at the end of the 19th century - 1.7, which was almost a record value in the entire history of Russia.

In the 1930s, the trend towards a decrease in generation replacement rates, caused by a decrease in the birth rate (the mortality situation practically did not improve), became predominant against the backdrop of fluctuations caused by the forced “building of socialism” and famine. The Second World War, in turn, increases fluctuations and causes another demographic crisis. The probability of surviving to the average age of motherhood again drops to 37%, and the birth rate - about 3 children per woman - turns out to be clearly insufficient for simple generation replacement (the maternal generation was replaced by a generation 44% smaller in number - the net reproduction rate population in the first half of the 1940s, according to our estimate, was 0.56). It is clear that if such a reproduction regime were maintained, the population would begin to decline rapidly in the future - at a rate of no less than 1.8% per year.

In the post-war years, the birth rate, after a short-term and insignificant compensatory growth, resumed its downward trend. At the same time, the two post-war decades were marked by a sharp decline in infant mortality - the chances of a girl becoming a mother quickly increased to 90-95% by the early 1960s. Thanks to this reduction in mortality, the reproduction regime in the 1950s - the first half of the 1960s still ensured simple replacement of generations (each new generation reproduced the parent one by 10-20 percent). However, even then the prospect of a transition to narrowed reproduction, when each new generation would be smaller in number than the parent one, became increasingly obvious.

Since the mid-1960s, the effect of reducing mortality has become insignificant. An increase in the probability of survival of a newborn girl to the average age of motherhood from 0.96 to 0.98 was not capable of seriously affecting the integral characteristics of population reproduction. The decisive factor in changes in reproduction rates in the last third of the 20th century and for the entire subsequent historical perspective is the birth rate. And it only for a short time, in the second half of the 1980s, rose to the level of 2.1 children per woman (the limit of simple reproduction at the current mortality rate). Therefore, it is not surprising that since the mid-1960s, a reproduction regime has been established in Russia that does not even ensure simple replacement of generations (“narrowed” reproduction). The fall in the birth rate in the 1990s further increased the degree of “underreproduction” (each new generation of children today is 30-40% smaller than their parents).

Since Russia's population has not been reproduced for four decades, the prospects for its growth due to natural growth in the next two decades are negligible. In the absence of additional migration and the birth rate maintaining the level of the second half of the 1990s, the population may decline at an annual rate reaching 1% per year, and, in the limit, up to 2% per year, as indicated by the natural increase rate stable population (20.3 per 1000 population), shown in Table 1.

With all the analytical value given in table. 1 and in Fig. 1 indicators, they are also not perfect. These indicators refer to the so-called “conditional” generations and represent, in essence, nothing more than an assessment of the actual demographic conditions of population reproduction in a given calendar year (and not a description of the actual progress of the reproduction process, as is often thought).

The quantitative characteristics of real population reproduction would correspond to these indicators only if these conditions remained unchanged for a sufficiently long time. But in reality they constantly fluctuate, and during the period of demographic transition they are subject to long-term and significant directional changes.

The popularity of indicators for conditional generations (“transverse” or transversal) is explained by the relative simplicity of their calculation. But it is possible to obtain a complete and deep understanding of what is actually happening with the reproduction of the population only when it is possible to use indicators for real generations, or cohorts (“longitudinal”, or longitudinal). It is these indicators, this time actually describing the real progress of the reproductive process, that are discussed in the subsequent sections of this article.

shows how many girls born to one woman in her lifetime, on average, will survive to the age of the mother at their birth, given the birth and death rates.

Excellent definition

Incomplete definition ↓

a general characteristic of the population reproduction regime, showing how many daughters a certain set of newborn girls will give birth to throughout their entire life ahead under a given fertility and mortality regime.

Excellent definition

Incomplete definition ↓

Net reproduction rate

a quantitative measure of the replacement of the mother generation by the daughter generation. It is calculated as the average number of daughters born to a woman in her entire life and surviving to the age of the mother at the time of their birth, given age-specific levels of fertility and mortality. The net population reproduction rate is equal to the gross population reproduction rate, adjusted using the numbers of survivors from the mortality table.

Excellent definition

Incomplete definition ↓

Net population reproduction rate

net population reproduction rate, Beck-Kuchinsky coefficient) is a quantitative measure of the replacement of the female generation, the generation of mothers, with the generation of daughters. The net population reproduction rate (Ro) occupies a central place in the system of population reproduction rates and is a general characteristic of the population reproduction regime. The idea of application and formula for calculating the net reproduction rate of the population was formulated by the German demographer and statistician R. Beck, and it was widely introduced into the practice of demographic analysis in the 1920-1930s by his student and follower, the German demographer and statistician R. Kuczynski and the American demographer and biologist A.J. Tray. At the same time, the French demographer P. Depois will propose to calculate the net population reproduction rate for real generations. The net population reproduction rate can be calculated for both the female and male population, but in the vast majority of cases it is used for the female population. It represents the average number of girls born to one woman in her lifetime who survive to the end of her reproductive period, given birth and death rates. This calculation formula is applied for one-year age intervals; if other intervals were used in the calculation (for example, 5-year), the resulting value must be multiplied by the appropriate value. In a simplified manner, the net population reproduction rate can be calculated using the formula: Ro = Rlx, where R is the gross population reproduction rate; lx is the number of women surviving to the average maternal age at childbirth, which ranges from 26 to 30 years. As a measure of the reproduction of a hypothetical generation, the net population reproduction rate is valid only for a stable population, that is, a population whose reproduction regime does not change over time. The size of such a population increases (decreases) by a factor of Ro over a time T equal to the average generation length. If Ro > 1, the population grows (expanded population reproduction; with Ro 1. O. ZAKHAROVA

Excellent definition

Incomplete definition ↓

NET REPLACEMENT RATIO OF POPULATION

NET RATIO OF POPULATION REPRODUCTION, net population reproduction rate, a quantitative measure of the replacement of the mother generation by the daughter generation, occupying the center. place in the system of population reproduction rates; a general description of the population reproduction regime, taking into account fertility and mortality. N.-k. V. n. (R0) is calculated separately for us. each gender. In the vast majority of cases, the net coefficient is used. reproducing women's stories about us. It represents cf. the number of girls born in a lifetime to one woman who survives to the end of the reproductive period at given levels of fertility and mortality:

where δ is the proportion of girls among newborns, x is age, f(x) is the age function of fertility, l(x) is the age function of woman survival, a and b are the boundaries of the reproductive period.

N.-k.'s calculations V. n. are performed according to the approximate formula:

where Fx is the same as f(x) on average for discrete age intervals from x to x + 1, i.e. age coefficients. fertility, Lx - avg. the number of living women according to the mortality table for the same intervals, and δ is taken to be independent of the age of the mother. Usually they deal with one-year intervals. If the values of Fx and Lx reduced to such an interval (i.e., to one year of age) are available only for n-year (for example, 5-year) age groups, then.

If the mortality table contains one-year Lx values, you can use their sums for each n-year interval:

Example of calculation of N.-k. V. n. based on Fx data for 5 year age groups of women for us. USSR in 1969-1970, see table.

Taking δ - 0.488 (see Sex ratio), we have R0 = 2.2815-0.488 = 1.113.

An approximate calculation of N.-k. is possible. V. n. using a simplified formula: , where R0 is the gross population reproduction rate, is the number of women surviving to the average age of the mother at the birth of children. This age varies little and is usually 28-30 years. If we take = 30, then for the given example R = 1.166, l30 = 0.954 (according to mortality tables 1968-71), R0 = 1.166*0.954 = 1.112.

Calculated for hypothetical generation, N.-k. V. n. the most complete interpretation is received within the framework of the model of reproduction of us, the regime of which does not change (stable population). Number such us. increases (or decreases) by R0 times during a time T equal to avg. generation length. If R0 > 1, num. us. grows (extended playback) if R00 = 1, number. us. does not change (simple reproduction).

In stable us. N.-k. V. n. associated with the true natural coefficient. growth of us. r by the ratio:

where e is the base of natural logarithms. In a real population, the reproduction modes of which are continuously changing, the relationship between population dynamics and the value of N.-to. V. n. is not so clear, because this dynamics also depends on the age structure of the population, which, in turn, determines the potential for population growth. If this potential is positive, then the number of us. can increase even when R00>.

The value of N.-k. V. n. to midday 19th century was exposed means. fluctuations, but, in contrast to the fertility and survival functions that determine this value, which reveal historical. a tendency towards directional changes, an average level around which the values fluctuated

N.-k. V. n., throughout history remained relatively stable and, as a rule, was close to the level of simple reproduction of us. (R0 = 1). For the initial phases of demographic transition is characterized by a temporary rise in N.-to. V. n., especially significant in developing countries in the 20th century. If in the 2nd half. 19th century in Western countries Europe, which was experiencing the early phases of the demographic revolution, had the highest values of N.-to. V. n. were ok. 1.5, then in the 2nd half. 20th century in some developing countries they reach 3.0 or more (one of the main manifestations of the demographic explosion). The difference in the meanings of N.-k. V. n. in modern world is large (see Population reproduction). The worldwide process of reducing N.-to. V. And. can also be traced in the USSR, where its value decreased from 1.680 in 1926-27 to 1.104 in 1975-76. At the same time, large differences in the size of N.-to remain. V. n. for the union republics.

For the first time he formulated the net coefficient. reproducing us. R. Beck. In practice demographic. analysis of N.-k. V. n. was widely introduced in the 20-30s. 20th century R. Kuchinsky and A.J. Lotka (Beck-Kuchinsky coefficient). At the same time the French scientist P. Depois proposed to calculate N.-k. V. n. for real generations. To assess the influence of the initial age structure of us. on coefficient reproduction in the USSR, an integral coefficient was proposed (1976). reproducing us. as Rs = R0 * VN, where VN is the net demographic potential. growth. Logical The development of this scheme is the introduction of the amendment of A. Ya. Kvasha, who proposed multiplying the demographic potential. growth is not ordinary, but so-called. cleared net coefficient L. Henri as the product of R0 and the ratio of the life expectancy of the generation of daughters (e´0) and the generation of mothers (e0). At the same time, the corrected N.-k. V. n. (Rk) has the form:

Rk = R0 * VN * e´0/e0.

Excellent definition

Incomplete definition ↓