Not an easy question to answer, the "bumper sticker" reaction to those that address the question will probably be yes. However for those that think beyond the bumper sticker that answer isn't as simple for a few reasons:
#1 In 1983 the law was changed. Prior to that there was the Civil Service Retirement System (CSRS) was phased out and replaced with the Federal Employees Retirement System (FERS). After that date new hires were not eligible for CSRS, they were automatically part of FERS. Existing CSRS employees were given the option of staying in CSRS or transitioning to FERS. Under CSRS, employees were not eligible for Social Security, however FERS employees do pay into Social Security. Since we are talking 40 years ago, the number of Federal employees still in CSRS is a very small portion of that population and it gets smaller by the day as those employees (if even still working) are transitioning to retirement now.
So in terms of "Federal" employees being in Social Security, they already are.
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#2 About 72% of State and Local employees are already part of Social Security. Of those that aren't 75% are concentrated in 8 states. The remainder are states where in general state and local authorities already participate in Social Security, but certain sections may not. Those that are often excluded include classes of employees grouped together as "First Responders". The reason is because of the physical demands of the job, they often work shorter careers and can retire earlier. Then probably going on to do other work. A similar concept with the military. I myself had a Naval career and moved on to a 2nd career in information systems. Neighbor across the street was a State Policeman, before retiring he started a small business and upon retiring it became his full time occupation.
As a general concept do you really want 67 police, fire fighters, and paramedics being the responders to crimes, fires, and natural disasters? Where these people instead of retiring at 40'ish after 20 years are just trying to hold on to full retirement age of 67? To me that doesn't make sense.
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#3 The budget impact projection (included in the literature below) show that mandatory conversion to 100% of state/local workers into Social Security would actually make things worse. Remember, because they aren't in the Social Security program, they are not eligible to receive benefits for employment in that job.
It shows that in the short term inclusion of those currently under an alternative/equivalent retirement plan would result in a short term 5% increase in funding, but that funding disappears and it would actually have a negative impact as they retire. Why? Because short term revenues would increase, but so would benefits - however they have not been paying into Social Security for decades meaning they were not contributing to the Social Security Trust Fund which was supposed to get us through the baby boomer bulge. Because they weren't paying into the "reserve" trust, ALL of their benefits would have to be paid from current revenues.
So mandatory conversion of state/local employees does not improve the social security situation.
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#4 The state/local governments that don't cover certain workers under Social Security are required by law to provide benefits at least as good as Social Security. Where states/local governments have opted out, they have a structure in place to pay those level of benefits. Where state/local governments opted into Social Security, they may still have a retirement system in addition to Social Security, but typically those benefits are designed to work with Social Security to provide for retirement income.
What I mean by that is in the opt out states, ALL retirement income comes from the fund. In states like Virginia, you pay part of each paycheck into VRS to fund retirement and pay Social Security. Your VRS is lower as a result. I'm looking at retirement in a year or two and my combined VRS + Social Security will replace about 78% of my working income. A decease in gross income sure, but one that is manageable with other things that change in retirement.
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#5 Then finally (for this post) you have answer the question of "how to transition to mandatory 100% participation?" Just like with the transition from CSRS to FERS the only reasonable way to do it would be to phase it in over time based on new hires and where existing/older employees are given the option to transition or remain with their current plan.
You'd be thinking of time horizons measure in decades (probably 4-6).
In addition, state/local legislative bodies would need to totally revamp their pension programs to align with the change in concept from "Our programs provides full retirement benefits without Social Security to we want the same level of coverage so we have to reduce our benefits to work in partnership with Social Security". That would be a nightmare.
WW