[Paper Review] From Ponzi Schemes to Benign Investment Dynamics: modelling Collapse, Stability, and a Path to Sustainability

Bernhard R. Parodi|arXiv (Cornell University)|Feb 21, 2026

Economic theories and models0 citations

TL;DR

The paper presents three discrete-time investment models with closed-form solutions (geometric, quasi-logistic, and SIR-based) that unify Ponzi-like collapse and benign, finite-horizon investment dynamics under a common capital-budget identity.

ABSTRACT

The population and capital dynamics of three stylized investment systems are mathematically described using discrete-time difference equations with closed-form solutions. The models share a common capital budget equation but differ in their demographic laws, which are geometric, quasi-logistic, or epidemiologic (SIR-based). The quasi-logistic model is designed as an analytically tractable non-Ponzi investment system: it generalizes the geometric model (and, in the limit of a constant growth rate, reproduces classical Ponzi dynamics) while closely mirroring the behaviour of an SIR-based model with decreasing effective growth. In all cases, promised returns are modeled as fixed per-period payouts on initial investment with principal repaid upon exit, so that aggregate liabilities depend only on the current number of active investors. Within this unified framework, classical Ponzi schemes arise as special cases that inevitably collapse, while suitable parameter choices in the quasi-logistic and SIR-based versions generate finite-horizon, legally benign "no-Ponzi game" investment schemes with analytically transparent conditions for collapse, stability, and sustained operation.

Motivation & Objective

• Motivate a unified mathematical framework for investment schemes that can exhibit both collapse and sustainability.
• Develop three demographic models (geometric, quasi-logistic, SIR-based) with closed-form capital dynamics.
• Identify parameter regimes that determine collapse versus viability and outline conditions for benign operation.
• Provide analytic insights into how lock-up periods and finite investor pools influence long-term outcomes.

Proposed method

• Define a common capital-budget identity with fixed per-period payouts and principal repayment.
• Derive closed-form capital dynamics for three demographic laws: geometric growth, quasi-logistic growth, and SIR-based growth.
• Introduce a novel quasi-logistic demography that yields a finite-horizon, non-Ponzi regime.
• Present exact/explicit solutions for discrete-time models and compare with SIR variants.
• Analyze how parameters like initial capital, lock-up time, and demographic rate affect stability and collapse.

Figure 1: Growth rates $n_{t}$ for the quasi-logistic growth model and for a couple of SIR-models. (For better visualization, only the interpolation lines connecting the discrete data points will be shown in this figure and in all other figures.) Top panel: Sigmoidally decreasing growth rates $n_{t}

Experimental results

Research questions

• RQ1 Under what parameter conditions do Ponzi-like schemes collapse versus persist or become benign?
• RQ2 How do different investment-demography laws (geometric, quasi-logistic, SIR-based) influence capital dynamics and sustainability?
• RQ3 Can a quasi-logistic or SIR-based framework produce finite-horizon, legally benign investment schemes within a unified model?
• RQ4 What role do lock-up periods and finite investor pools play in transitions between collapse and viability?

Key findings

• Three discrete-time models with closed-form solutions describe capital dynamics under different demographies.
• Ponzi-type collapse emerges endogenously under adverse parameters, while quasi-logistic and SIR-based models can produce finite-horizon benign schemes.
• The quasi-logistic model embeds geometric growth as a special case and mirrors SIR-like decreasing growth with a sigmoidal investor base.
• A unified framework links classical Ponzi schemes, fragile systems, and benign pooled-income products via a common budget identity.
• Critical parameter regimes delineate collapse from stability and indicate a path toward sustainable investment dynamics.
• The paper proposes a novel temporary instrument viable under controlled conditions.

Figure 2: Demographic development for the quasi-logistic model. Developments are shown under different lock-up period conditions: without exits from the system (lock-up period $T=200$ , representing $T\rightarrow\infty$ , dashed lines) and with exits after $T$ = 30 periods of participation (solid li