A few readers have requested some additional explanation of the equivalent distribution simplification that I used in my previous data science article about Social Security. That article relies on a present value comparison between total benefits received and total contributions made for the cohort of workers who reach full retirement age in any given year.

Tracking the benefits for a cohort presents several problems: i) payments have to be followed years into the future; ii) the Social Security Administration (SSA) doesn't really publish enough detail in order to properly tease out the benefits that are unique to a given cohort; iii) accurate handling of dependent (spouse, children, parents, etc.) benefits can be even more complicated than that for workers. To skirt these problems I recognized and used an equivalent distribution simplification in which total benefits paid in any year are a very good statistical representation of the total payments which would be made to that year's retirement cohort over time.

Table I here shows schematically how/why the equivalent distribution simplification works.

The leftmost column in Table I designates different years before and after the year a cohort of workers reach full retirement age. To keep the explanation abstract I have used 0 rather than a specific year for the retirement year, and I have designated years before and after the retirement year with an appropriate integer (plus or minus).

The two columns left of the vertical bars in column E represent the benefits paid to cohort 0 workers and dependents in different years. Payments to workers are designated with a W, and payments to dependents are designated with D. Cells with payments to dependents are also represented with gray background shading. The index after the W or D designates the year relative to full retirement year. So D-3 means payments to dependents three years prior to the full retirement year. The age of the associated workers in the cohort is indicated after the capital A, The final Y and an integer (plus or minus) designate the actual year relative to the retirement year for the cohort.

I have used color coding to signal the correspondence between the payments to cohort 0 on the left and payments to a corresponding and otherwise equivalent different cohort on the right. For example payments to workers (and their dependents) who are retiring at age 62 three years before full retirement age are found in cells C5 and C6 on the left and F8 and F9 on the right. We should expect these benefit payment amounts to be statistically equivalent even though the cohorts differ by three years. Working down columns C and D and across rows 8 and 9 to the right, everything which needs to be tracked for cohort 0 in different years corresponds to something found in rows 8 and 9 for the total benefit payments in year 0. The statistical equivalences work because the numbers of workers and dependents are large enough, and any demographic (age/sex/population) changes occur slowly.

This equivalent distribution simplification should be accurate enough for the trends that are demonstrated in the previous blog article. SSA has access to demographic information about its worker retirees and dependents which could be used to further improve the quality of the equivalent distribution simplification for routine use in correct balancing of income contributions and benefits. If very long retirement periods (30 to 40 years) were to develop, there might be more serious inaccuracies in the equivalent distribution simplification, but very long retirement periods likely cause other much more significant problems for Social Security in general.