Steve Keen and Matheus Grasselli et. al. have developed a simple model of debt-deflation. Here is the pdf of Grasselli and Costa Lima’s article. The work is a mathematical formulation of Hyman Minsky’s Financial Instability Hypothesis
The basic system consist of four differential equations:
\[\dot{\omega} = \omega [ \Phi (\lambda) - \alpha ]\] \[\dot{\lambda} = \lambda [ \frac{ \kappa ( 1 - \omega - r d )}{ \nu } - \alpha - \beta - \delta ]\] \[\dot{d} = d [ r - \frac{ \kappa ( 1 - \omega - r d )}{ \nu } + \delta ] + \kappa ( 1 - \omega - r d ) - (1 - \omega) + p\] \[\dot{p} = p[\Psi(\frac{\kappa(1-\omega - r d )}{\nu } -\delta) - \frac{\kappa(1-\omega - r d )}{\nu}+\delta]\]The variables involved are:
- \(\omega\) - wage share
- \(\lambda\) - employment rate
- \(d\) - debt
- \(p\) - ponzi speculation
\(\Phi (\lambda)\) is the well known Phillips curve, here specified as
\[\Phi(\lambda) = \frac{\phi_1}{(1 - \lambda)^2} - \phi_0\]where \(\phi_0\) are \(\phi_1\) are constants. This specification allows for a non-linar relationship between the wage share and the employment rate. As \(\lambda\) moves towards full employment more than proportional increases in \(\Phi\) accelerates the relative increase in the wage share \(\omega\) through equation (1). \(\alpha\) is the growth rate of labour productivity.
\(\kappa(1 - \omega - r d)\) is the investment function. \((1 - \omega)\) being the gross profit share and \(r d\) the costs of servicing debt \(d\) at interest rate \(r\), here specified as
\[\kappa(1 - \omega - r d) = \kappa_0 + \kappa_1 e^{\kappa_2(1 - \omega - r d) }\]with constants \(\kappa_0, \kappa_1\) and \(\kappa_2\), so that investments are an increasing function of profits. Dividing investments by the capital-output ratio \(\nu\) gives the increased production. To arrive at the change in the employment rate \(\lambda\) in equation (2) we have to subtract labour productivity increase \(\alpha\), growth of work force \(\beta\) and depriciation \(\delta\).
The growth rate of the economy becomes:
\[\frac{ \kappa ( 1 - \omega - r d )}{ \nu } - \delta\]so that existing debt in equation (3) grows by
\[d [ r - \frac{ \kappa ( 1 - \omega - r d )}{ \nu } + \delta ]\]pluss investments
\[\kappa ( 1 - \omega - r d )\]minus profits
\[(1 - \omega)\]pluss ponzi speculation \(p\) driven by the growth rate of the economy as in (4).
The speculation function
\[\Psi(\frac{\kappa(1-\omega - r d )}{\nu } -\delta)\]is here specified as
\[\Psi(1 - \omega - r d) = \Psi_0 + \Psi_1 e^{\Psi_2(1 - \omega - r d) \ }\]with constants \(\Psi_0, \Psi_1\) and \(\Psi_2\) so that speculation increases with the growth rate.
The C++ code can be found here.