Projeção da população do Brasil : aplicação do método cadeia de Markov


  • M. J. McCullagh
  • Speridião Faissol
  • John Paul Cole


Geografia da população, Markov, Processos de, Previsão demográfica, Geografia quantitativa


The article employs a technique familiar in statistics (Markov Chain), to project the growth of the Brazilian population and the internal migrations, over a long period, up to even 100 years, an extremely laborious piece of work but which is facilitated by the use of the calculating capacities of a computer (a few minutes only for a large-size computer). The method is still a long way from being used with full reliance because there are still a number of details that require better knowledge; since however acquiring this knowledge is a lengthy process, the Department is now publishing not only the method but also some of the results obtained so that, in the light of such results, criticism may be forthcoming enabling us to move step by step with a view to improvement.

            The method is in essence very simple inasmuch as it is made up of a series of calculations in chain, (following indexes which constitute the data of entry in the program) with regard to how many persons are born, how many die and how many migrate from one area to another. As the per capita income is regarded as a constituent that simultaneously affects that birth-rate, the death-rate and the migration, the program takes into account also the per capita income for each region and, by extension, the national income.

            In resume, the program furnishes a model of income growth, of population and its mobility, thus representing a. real model of the Brazilian spatial development. As for every kind of model it assumes a certain number of postulations and its results are valid only in ratio to these premises.



            The first among the premises of the program is that the per capita income is a determining factor in relation to birth-rate, death-rate and migrations.

  1. a) The birth-rate is high where there is a low per capita income, increases slightly with the increase of income in consequence of the improved health conditions of the female population between 15 and 49 years of age, and begins to decline when the per capita income reaches 600 dollars, stabilizing itself when it reaches the 900 dollars level.
  2. b) The death-rate is also high and only tends to decrease at the level of the 300 dollars status of income, lessening to half when the income reaches 600 dollars and to a third when it attains 900 dollars, also reaching stability at this point.
  3. c) The migrations proceed from areas of lesser income to areas of higher income, in direct ratio to the income differential and inverse to the distance. As the model is determinant there is no prevision for return migration, although restriction on migration has been taken into account in the form of a percentage of migrants per annum. Another aspect of the migratory problem is that of its selectivity; the model adopts a premise at this point, namely the propensity to migrate defined by age groups, it being move intense in the 20 to 40 years age group and less just as much in the groups above as in those below that age bracket.



  1. In the first place the model requires utilization of population data, by age groups and by spatial units. The model allows for up to 90 groups (the model developed used 17 groups of 5-year span) and 40 spatial units (in the example the five great Brazilian regions were utilized: North, North-East, South-East, South and Center-West).
  2. The second element is the total income (the value in dollars was employed for facility of comparison, the cruzeiro value having been converted into dollars at the 1966 rate, namely 2. 2 cruzeiros per dollar).
  3. The third refers to the birth-rate, with application here of the fecundity index modified by income.
  4. The death-rate, by age groups, is the element of data that follows. Obviously no data of this nature exists for Brazil. Approximate indexes were elaborated, consistent with the theoretical concepts on the death-rate, employing certain figures of the infantile and the gross death-rates for development of a table showing a high death-rate in the 0-year to the 4-year-old bracket (corresponding to 130 per mil), lessening in the following brackets and increasing again after 40 years of age, until completion of the cycle at 85 years, the point at which the index equals a thousand. per mil, in other words 100%.
  5. The income for each region and, by extension, the national income, is the fifth data-element utilized and, to this income is applied a rate of growth which, in the example, was 7% per annum ( 40% for a 5-year period) and a deflating coefficient for this rate Which can keep the rhythm unchanged if equal to 1, increase it if over 1 and reduce it if lower than 1. In the example the cadence decreases slowly from 7% to 5% per annum at the end of 50 years.

            The growth-rate may vary with each unit, but in the example the same rate was used for each of the five regions.

  1. Inasmuch as the migrations occur in inverse ratio to the distances, the model includes figures relating to the distances in straight-line between the most important capitals of each region, as for instance, between Rio and São Paulo and between Belem and Recife. The model also permits utilization of an exponent for modifying these distances but, in the example, this was not used.
  2. In relation to the migrations, the following premises were furthermore applied:
  3. a) Only 0,5% of the population migrates annually. This restrictions became necessary because an unrestricted flux of migrants, due only to income differential, would cause the whole population to be concentrated in São Paulo at the end of little more than one century.
  4. b) Because the program adopts the premise of selective migration, an index of propensity to migrate was developed, consistent with the principle that persons in the 20 to 30 years-of-age group have a greater tendency to migrate.
  5. c) Insofar as income is concerned, the data-element used in the program derives from the premise that the migrant takes along with him, his capacity to create revenue, to the region of his destination.



  1. The Brazilian revenue in 1966, converted in to dollars at the rate of 2. 2 cruzeiros per dollar, gave an approximate total of $200,000,000 or 230 dollars per capita for a population of almost 86 million people.
  2. In the year 2001 with a total income-growth of 7% per annum (with, however, a slow decrease in the rhythm) the per capita income would surpass 1,060 dollars and the population would reach the 180 million mark.
  3. The Southern Region whose income in 1966 was of 325 dollars, would surpass 1,270 dollars in in come, namely almost 6 times as much. The population which was of 32 million would increase to almost 90 million, or in other words, the percentage of its participation in the Brazilian total would grow from 36% to 50%. In the North-East income would progress from the present 110 dollars to 550 dollars, an increase of 5 times as much, and the population would pass from 18 million to almost 27 million 1nhabitants.
  4. The population of O to 19 years of age, which today represents about 43% of the total, would become 32% in the year 2001; the group 20 to 40 years of age would remain steady at the Present 32% to 32,5%, but the group above this would increase from 25% to almost 35%, with all the consequences resulting from obviously very different occupational structures.

            The model supplies also the following information:

  1. The total population and for each region, by age groups.
  2. The number of births and deaths.
  3. The total national revenue and the per capita income for each region.
  4. Migrations from each region to each other region.