Burying Samuelson’s Multiplier-Accelerator and Resurrecting Goodwin’s Growth Cycle in Minsky

Abstract

Samuelson’s Multiplier-Accelerator model is based on the economic mistake of adding together desired investment and actual savings to derive aggregate expenditure, when it is the sum of actual investment and actual savings. Its fluctuations are not a model of the business cycle, but convergence to or divergence from the trivial solution. Goodwin’s relatively neglected growth cycle model is shown to be a valid foundation for modelling the business cycle and is derived directly from macroeconomic definitions. When the private debt ratio is added to those definitions, Minsky’s “Financial Instability Hypothesis” results. I illustrate modelling of this hypothesis in the open-source system dynamics program named after Minsky, which has been designed predominantly to enable monetary dynamics to be modelled. I use Minsky’s unique feature of Godley Tables to show the macroeconomic difference between the false Loanable Funds model of banking and the realistic “bank originated money and debt” model. I show that a third-order difference equation growth cycle model can be derived from the concept of an investment accelerator. I close with an Appendix showing the use of Godley Tables in epidemic modelling.

Appendix: Using a Godley Table in Epidemiology

The basic SIR model of susceptibility, infection and recovery (Kermack, McKendrick et al., 1927 [1997]) can be seen as an extension of the predator–prey model, in which the consequence of predation is not death but infection.⁷ In the basic predator–prey model, an assumed exponential growth of the prey population x at a rate $\frac{1}{x}\frac{{\text{d}}x}{{{\text{d}}t}} = \alpha$ is reduced by a constant $\beta$ times y, the prey population, so that $\frac{1}{x}\frac{{\text{d}}x}{{{\text{d}}t}} = \alpha - \beta \times y$. Interaction between the predator and prey is thus shown as a multiplicative relationship.

In modelling a pandemic, the population N is normally treated as a constant, since it changes far less rapidly than the epidemic spreads. The rate of change of the fraction of the population that is infected depends on the interactions of those infected I with those susceptible S, which in turn depends on the frequency of both groups in the overall population, S/N and I/N, and the transmissibility of the disease, which is modelled by the parameter $\beta$:

$$\frac{{\text{d}}}{{\text{d}}t}\frac{S}{N} = - \beta \cdot \frac{I}{N} \cdot \frac{S}{N}$$

(18.32)

Since population is treated as constant, this reduces to:

$$\frac{{\text{d}}}{{\text{d}}t}S = - \beta \cdot \frac{I \cdot S}{N}$$

(18.33)

Since the increase in those infected is equal to the fall in those who are susceptible, the rate of growth of those infected is the negative of the rate of decline of those susceptible, minus the recovery rate R, which is modelled as a parameter $\gamma$ times the number infected:

$$\begin{aligned} & \frac{{\text{d}}}{{\text{d}}t}I = \beta \cdot \frac{S \cdot I}{N} - \gamma \cdot I \\ & \frac{{\text{d}}}{{\text{d}}t}R = \gamma \cdot I \\ \end{aligned}$$

(18.34)

Figure 18.10 shows this model implemented in Minsky, using time constants rather than parameters. The Godley Table’s tabular format makes it easy to see the interrelations between the susceptible, infected, and recovered compartments. Flowchart tools are only needed to define the flows themselves (and future versions will allow these to be defined off the canvas, using standard L^AT_EX equation notation).

Fig. 18.10

Cyclical growth in the third-order Multiplier-Accelerator Model, with a higher savings rate meaning lower growth

The Godley Table interface also makes it very easy to extend this model to a more realistic situation in which there is a more complicated transmission chain—see Fig. 18.11. Other comparmentalizations—such as dividing the susceptible population into the general public and medical staff, including quarantined versus non-quarantined, hospitalized versus non-hospitalized, etc.—are equally straightforward to add and define by adding additional columns and matching flows to Fig. 18.12.

Fig. 18.11

Simple SIR model of a pandemic

Fig. 18.12

SEIRD model developed by editing Godley table of SIR model