This paper follows earlier ones by the same authors in applying mechanism design to finding an optimal non-linear pension-benefit rule for flexible retirement. It is assumed that individuals have private information about their expected life spans. The government's goal is to design a pension system (a payroll tax and a function that relates benefits to length of employment) that maximizes a social-welfare function and satisfies a social budget constraint (without satisfying the individual ones) redistribution. Since individuals with different life expectancies optimize their employment lengths conditionally on the benefit function, the government must also take into account incentive constraints. The authors' former studies reduced the inflexibility of the optimum and the excess in redistribution by rendering the social-welfare function concave. The new findings in this paper are: (1) some redistribution is inevitable in any reasonable pension system, and (2) second-best solutions may be local indeterminate, while (3) returning to utilitarianism, the minimization of redistribution is considered.