The model.
Consider an individual at the beginning of year t, who has not yet retired.
Looking ahead, he will receive wage income in year a as long aa he continues to work; if he is retired in year a,
he will receive real retirement benefits 8.
(We adopt the convention that if a is the first calendar year during which the person has no wage earnings,
he is assumed to
have left the firm during the previous year, at the age that he was on January 1 of year a.) Let r denote the first full year of the individual's retirement
(that is, the first year in which the individual has no wage earnings).
As described above, these benefits will depend on the person's age and years of
service at retirement, and on his earnings history; thus we typically write