Why Y chromosome is shorter and women live longer?
  We have used the Penna ageing model to analyze how the differences in evolution of sex chromosomes depend on the strategy of reproduction. In panmictic populations, when females (XX) can freely choose the male partner (XY) for reproduction from the whole population, the Y chromosome accumulates defects and eventually the only information it brings is a male sex determination. As a result of shrinking Y chromosome the males become hemizygous in respect to the X chromosome content and are characterized by higher mortality, observed also in the human populations. If it is assumed in the model that the presence of the male is indispensable at least during the pregnancy of his female partner and he cannot be seduced by another female at least during the one reproduction cycle - the Y chromosome preserves its content, does not shrink and the lifespan of females and males is the same. Thus, Y chromosome shrinks not because of existing in one copy, without the possibility of recombination, but because it stays under weaker selection pressure; in panmictic populations without the necessity of being faithful, a considerable fraction of males is dispensable and they can be eliminated from the population without reducing its reproduction potential. 
Przemyslaw Biecek, Stanislaw Cebrat