The mandatory pension pillar is usually supplemented by a voluntary one. In our simple model, voluntary pensions partly replace mandatory ones without affecting the outcomes: the voluntary pensions are indifferent. This result may serve as benchmark.

When individuals underreport their incomes, they take into account their private gains and moral losses, the latter depending on the acquaintances’ previous underreports. We prove that under quite natural assumptions the process globally converges to the symmetric steady state.