Written by Steve Keen, Steve Keen’s Debtwatch

Minsky’s Financial Instability Hypothesis (FIH) is an emergent property of macroeconomic models derived directly from macroeconomic definitions.

This is Part 5 of a paper presented at the International Conference Minsky at 100 Revisiting Financial Instability, December 16-17 2019 – Universita Cattolica del Sacro Cuore Milano.

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This paper is posted in five parts:

Part 1: Deriving a Minsky Model

Part 2: Simulating Loanable Funds and BOMD

Part 3: Accounting For The Great Moderation & The Great Recession

Part 4: Nonlinearity and Realism

Part 5: Appendix and References (this article)

Appendix

The basic linear Minsky model is:

Where the following shorthand expressions are used:

Spelling out these shorthand expressions yields the fully specified model, which makes it easier to identify the nonlinear feedbacks in this model. Variables that interact nonlinearly with other variables in this system are highlighted in red: there are two dampening nonlinear feedbacks in the equation for λ, one amplifying feedback for ω, and two amplifying feedbacks for d (including one term in d-squared):

The “good” equilibrium of this model can be derived by solving for the zeros of these equations via the substitution that

yields:

This equilibrium is in terms of the profit share, employment rate and debt ratio: the wages share is a derivative of these, since

This residual role for the wages-share of output manifests itself in the model dynamics as well: before the crisis, the wages share falls as the debt level rises, while the profit share fluctuates around its equilibrium. This confirms Marx’s intuition in Capital I that wages are a dependent variable in capitalism:

“To put it mathematically: the rate of accumulation is the independent, not the dependent, variable; the rate of wages, the dependent, not the independent, variable” ( – – Marx 1867, Chapter 25, Section 1)

Loanable Funds & BOMD

The key differential equations for the models of Loanable Funds and BOMD as shown in Equations (1.6) and (1.8) respectively. The definitions they share are shown in Equation (1.25):

Deriving a Price Equation

This section derives a pricing equation similar to Kalecki’s markup pricing equation ( – – Kalecki 1938; – – Kalecki 1971; – – Kriesler 1988) from a simple monetary model of circulation. Price P is treated as an equilibrating function driven by the difference between the monetary value of demand D$ and the monetary value of output S$, where the rate of convergence is given by the time constant τp . Then we have

The monetary value of output S$ is price P times physical output Q. Physical output divided by the output to labour ratio a determines employment L, so that we can write

In this monetary model, employment is determined by the wage bill W divided by the money wage w$. The division of income between workers and capitalists is given by 0 > s > 1, with s x D$ going to capitalists as gross profits, while (1 – s) x D$ goes to workers as wages. Thus given:

We can write S$ as:

At the equilibrium price PE, the monetary value of supply equals that of demand, so that:

Solving for PE yields:

This can be rewritten in terms of the wages share of income ω:

Therefore:

This is equivalent to Kalecki’s markup pricing equation, with the markup being 1 divided by the workers share of income. We can now rewrite the inflation equation (1.26) in terms of the wages share of income: