For the purpose of constructing a model which excludes the fact that the number of workers (who pay the bill) is less than the number of retirees (who collect via SSA from the workers), consider the following:
Start with an eighteen-year old entering the workforce today earning a $10,712 annual minimum wage until retirement
at age sixty-seven. 10.6% of his wages would be paid into payroll taxes for the retirement portion of Social Security ($1,135/year). If 100% of his previously forced payments were instead paid into a private account at a 6% constant-dollar rate of return (which is below the historical norm) then this yields a $359,406 retirement fund.
If we apply the government's Thrift Savings Plan calculator http://www.tsp.gov.with its historically low 4.25% annuity interest rate, the fund can buy a lifetime annuity paying $2,073/month, with COLA, which is more than twice what the worker earned while working.
On the other hand, Social Security promises about $590/month, 72% less than the private example. This individual would only get $413 per month - 20% of the private annuity.
So, Social Security is actually detrimental to the very people it purports to help. The Social Security Act was made a law on the basis that it would provide a safety net to those who are too old or too sick to work. But, as demonstrated above, the opposite is true; it cripples the very people it is supposed to help.