PROPOSITION 1: A Ponzi scheme may exist if an economy has a large public sector (” is bounded below), and the assets of the state could be used for a bailout ($ is bounded below), and the probability of early termination of the Ponzi scheme by a regulator is low (2 is bounded above), and there is inexpensive access to citizens through the mass-media (c is bounded above), and there are no severe penalties on promoters of Ponzi schemes (d is bounded above).

PROPOSITION 2: A Ponzi scheme will exist even under partial bailout, if the condition in Proposition 1 holds and the probability of bailout, $, is higher than (1-n*), where n* is the critical fraction of citizens that are required to be involved for there to exist the possibility of a bailout.

—- Utpal Bhattacharya, “The Optimal Design of Ponzi Schemes in Finite Economies”, 2003.