The mathematics of Ponzi schemes

A first order linear differential equation is used to describe the dynamics of an investment fund that promises more than it can deliver, also known as a Ponzi scheme. The model is based on a promised, unrealistic interest rate; on the actual, realized nominal interest rate; on the rate at which new deposits are accumulated and on the withdrawal rate. Conditions on these parameters are given for the fund to be solvent or to collapse. The model is fitted to data available on Charles Ponzi's 1920 eponymous scheme and illustrated with a philanthropic version of the scheme.Ponzi scheme; differential equation; market; bond

The Discrete Nerlove-Arrow Model: Explicit Solutions

The Discrete Nerlove-Arrow Model: Explicit Solutions

On the local stability of nonautonomous difference equations in Rn

We prove several theorems on the local stability of an iterative process associated with a nonautonomous difference equation in Rn. These results provide general conditions under which the common fixed point X* of a family of operators is uniformly stable, uniformly attractive, or uniformly exponentially stable. The stability conditions are obtained by majorizing products of Jacobian matrices in a neighborhood of X*. When the Frechet derivatives are equicontinuous at X*, majorizations at X* suflice to ensure stable behavior. Nonuniform stability conditions are discussed. Stability conditions are also investigated when the spectral radii of the Jacobian matrices at X* are uniformly bounded below 1

Un modèle démo-économique de la Révolution Industrielle

This paper describes a two-sector demo-economic model (agricultural and non-agricultural sectors) applied to Europe and spanning the period from the neolithic agricultural revolution to the Industrial Revolution. The model describes the "incessant contest" between population growth and food production. As long as per capita agricultural output is above a critical minimum, the population is assumed to grow at a constant rate. When this output drops below the minimum, the population is subjected to random mortality "shocks" which lower the population until the production grows above the minimum. Society is thus in a "Malthusian trap". The average magnitude of the mortality crises is assumed to decrease as capital increases, which captures an increased "resistance" that comes with increased knowledge and technology. The slow accumulation of capital thus diminishes the severity of the mortality shocks; as a result both the population of the non-agricultural sector and capital grow sufficiently to bring about a permanent escape from the Malthusian trap, i.e. the Industrial Revolution

Un modèle démo-économique de la Révolution Industrielle

This paper describes a two-sector demo-economic model (agricultural and non-agricultural sectors) applied to Europe and spanning the period from the neolithic agricultural revolution to the Industrial Revolution. The model describes the "incessant contest" between population growth and food production. As long as per capita agricultural output is above a critical minimum, the population is assumed to grow at a constant rate. When this output drops below the minimum, the population is subjected to random mortality "shocks" which lower the population until the production grows above the minimum. Society is thus in a "Malthusian trap". The average magnitude of the mortality crises is assumed to decrease as capital increases, which captures an increased "resistance" that comes with increased knowledge and technology. The slow accumulation of capital thus diminishes the severity of the mortality shocks; as a result both the population of the non-agricultural sector and capital grow sufficiently to bring about a permanent escape from the Malthusian trap, i.e. the Industrial Revolution.Industrial Revolution; Malthusian Trap

The mathematics of Ponzi schemes

A first order linear differential equation is used to describe the dynamics of an investment fund that promises more than it can deliver, also known as a Ponzi scheme. The model is based on a promised, unrealistic interest rate; on the actual, realized nominal interest rate; on the rate at which new deposits are accumulated and on the withdrawal rate. Conditions on these parameters are given for the fund to be solvent or to collapse. The model is fitted to data available on Charles Ponzi's 1920 eponymous scheme and illustrated with a philanthropic version of the scheme

The debt trap: a two-compartment train wreck

We aim to shed light on the debate among policy-makers trying to find prescriptions that will take troubled economies out of their debt trap. We do this with a highly stylized two-compartment dynamic model consisting of the stocks of money in Government and Society. The dynamics of the system are described by a simple four-parameter linear system of two differential equations. The solutions are investigated in closed form and provide precisely quantified "escape conditions" from the debt trap: receipts must be slightly larger than outlays and there must be sufficient annual inflows of funds into the system. The model fits the data for the U.S. between 1981 and 2012 with a coefficient of correlation of 0.996. The model is used to extrapolate the two stocks beyond 2012 with three escape scenarios which shed light on monetary flows needed to take the U.S. economy out of its debt trap