Consider that the population of a given country is classified in household terms into the following three classes of households: 1) High-Income (A); 2) Middle Income (M); 3) Low Income (B). Also consider that in a study that involved surveys of several ten-year population censuses, it was estimated that between periods of one decade to another, the following changes occur, on average, in the structure of the distribution of family income in this economy:
• Of the high-income families (A), 10% decrease their income and move to the middle-income class (M);
• Of the middle-income families (M), 10% manage to ascend to the high-income class (A), but the greater proportion of 20% lose income, falling to the low-income class (B);
• Of low-income families (B), a significant portion of 30% of them manage to ascend to the middle-income class (M).
Assuming that these percentages of changes in families between income classes still tend to last a long time in the future, and that in the last population census it was obtained that of the total families, 10% belong to class A, 40% to class M and 50% to class B, build a Markov-type model to determine:

1. The equations that describe the projections of the trajectories of the proportions of families per decade into the future;
2. Based on these equations, is it possible to say that, in the long run, the proportions of families tend to a fixed distribution of family income?

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1 P A t 1 P A t 0 1 P A t P M t 1 P M t 0 1 P A t 0 2 P M t 0 3 P B t P B t ... View the full answer