Debt, Inequality and Crisis
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, Ch. 25 S. 1)
Talk about centralisation! The credit system, which has its focus in the so-called
national banks and the big money-lenders and usurers surrounding them, constitutes
enormous centralisation, and gives this class of parasites the fabulous power, not
only to periodically despoil industrial capitalists, but also to interfere in actual
production in a most dangerous manner—and this gang knows nothing about
production and has nothing to do with it. (Marx 1894, Ch. 33)
The alternative polar view, which I call unreconstructed Keynesian, is that capitalism
is inherently flawed, being prone to booms, crises, and depressions. This instability,
in my view, is due to characteristics the financial system must possess if it is to be
consistent with full-blown capitalism. Such a financial system will be capable of both
generating signals that induce an accelerating desire to invest and of financing that
accelerating investment. (Minsky 1982, pp. 279-80)
From the perspective of economic theory and policy, this vision of a capitalist
economy with finance requires us to go beyond that habit of mind which Keynes
described so well, the excessive reliance on the (stable) recent past as a guide to the
future. The chaotic dynamics explored in this paper should warn us against accepting
a period of relative tranquility in a capitalist economy as anything other than a lull
before the storm. (Keen 1995, p. 634)
Introduction
When the economic history of our Millennial epoch is written, three stylized facts will dominate: rising
inequality, rising private debt, and a sudden transition from apparent tranquillity to crisis. All three
stylized facts emerge from a model of Minsky’s Financial Instability Hypothesis developed before the
“Great Moderation” and the crisis (Minsky 1977, Keen 1995). The core equations of the model can be
derived from identities for employment, income distribution and debt to GDP ratio. When expressed in
a model with linear relations, the phenomena of the “Great Moderation” and the “Great Recession”,
rising inequality, rising private debt and a final crisis all emerge.
A set of identities
The dynamic identities from which the model can be derived are the following:
The employment rate will rise if economic growth exceeds the sum of population and labor productivity
growth;
The wages share of output will rise if money wage demands exceed the sum of labor productivity
growth and inflation; and
The private debt to GDP ratio will rise if the rate of growth of private debt exceeds the sum of the rate
of economic growth and inflation.
The macroeconomic behaviour these equations generate cannot be deduced from the statements or the
equations themselves, but are an emergent property of the system which, for a wide range of functional
forms and parameter values, follows the Pomeau-Manneville, inverse-tangent route to chaos (Pomeau
and Manneville 1980), with diminishing cycles in the first two variables followed by increasing cycles,
and then a crisis with the employment rate and wages share of employment collapsing to zero and the
private debt ratio heading to infinity.
Generic Dynamics
Equation (1.1) states these three relationships in generic form.1
  YR     
  w N    P 
(1.1)
d  D   YR  P 
Here  is the ratio of employment L to population N    L N , YR is the rate of real economic
growth,  is the rate of growth of labor productivity,  is the rate of population growth,
 is the
wages share of output   wR  L Y R , w N is the money wage, P is the inflation rate, and d is the ratio
of private debt D to nominal output  d  D YN  , and the dot notation represents percentage rates of
growth so that  
D
1 d
1 d
1 dd
1 dP
1 dYR
1 dwN
, YR 
,
, wN 
, d
, P
, and
 dt
 dt
d dt
P dt
wN dt
YR dt
1 dD
.
D dt
Stated in this form, the equations are simply identities. To convert them into a model, causal
relationships are needed for the rate of economic growth, the rate of change of money wages, the rate
of change of private debt, and the rate of inflation. I start with the simplest possible behavioural
relations and introduce more realistic relations later. The remarkable finding is that even the simplest
possible behavioural relations generate stylized behaviour which corresponds to recent economic
history:
Income inequality rises;
The private debt to GDP ratio rises; and
A crisis occurs.
1


Here the dot superscript represents a percentage rate of change  x 
values and N indicates nominal values.
1 dx 
  . The subscript R indicates real
x dt 
The simplest possible model
To convert these identities into a model, causal relations are needed to determine its four determinants:
the rate of economic growth, the rate of change of wages, the rate of change of private debt, and the
rate of inflation. Following Goodwin (Goodwin 1967) I use linear relationships between the real capital
stock KR and real output, and between real output and employment L, while the rate of change of wages
is modelled as a linear response to the employment rate. Extending Goodwin, the rate of investment is
modelled as a linear response to the rate of profit, with debt finance used when this results in a level of
desired investment that exceeds profits (in this simplest possible model, inflation is ignored by setting
P  0 , so that all variables are in real rather than nominal terms). Goodwin assumed that all profits
were invested, and ignored depreciation. I include a linear depreciation rate (KR).
YR  KR v
L  YR a
 L N
dwR
 wR  w fn   
dt
 R  YR  wR  L  r  D
s  1  r d
 r   R KR   s v
dKR
 IR   KR  KR
dt
IR  Ifn  r   YR
dD
 IR   R
dt
w fn     S     N 
Ifn  r    S   r   N 
(1.2)
Following Goodwin, I assume constant growth in labor productivity and population at the rates  and 
per year respectively:
d
a  a
dt
d
N   N
dt
This results in a three-system state reduced form model:
(1.3)
  I  


d
      fn r   K       


dt

 v

d
     w fn      
dt
(1.4)
 I  

d
d   Ifn  r    s   d   fn r   K 
dt
 v

Using the linear relations for investment and wage change functions, this yields:



 1  r d

S 
N 




d
v

       
   


K
dt
v









d
     S     N    
dt


 1  r d

S 
N 


d


v
 1  r d


  
d   S  
  N   1    r  d   d  
K
dt
v
v










(1.5)
Only one of the system’s three equilibria involve non-zero values for the employment rate and the
wages share of output:
e 
  S  N
S
r             KR      r  2
e 
v
 S     
2

de 
v   N      r   r       KR       
1
 
   

     KR 
       KR   N  v 2 

S


This main behavioural determinants of the stability of this equilibrium are the parameters for the
investment function pS and pN. For low values of pS and high values of pN, the system is stable and
displays the dynamic characteristics shown in Figure 1.
(1.6)
Figure 1: The model with stable values for pS and pN
Stable system (Linear functions)
Stable
For higher values of pS and lower values of pN, the equilibrium is unstable and the model’s dynamics are
as shown in Figure 2.
Figure 2: The model with unstable values for pS and pN
Unstable system (Linear functions)
Unstable
The unstable outcome is the one of interest here, since—with limitations due to the absence of
inflationary dynamics and the use of linear functions, which are addressed in the next two sections—this
outcome resembles the actual historical record: a period of diminishing cycles in employment and
inflation2 in the context of a rising debt to GDP ratio gave way to a period of rising cycles and ultimately
crisis.
Goodwin, R. M. (1967). A growth cycle. Socialism, Capitalism and Economic Growth. C. H.
Feinstein. Cambridge, Cambridge University Press: 54-58.
Keen, S. (1995). "Finance and Economic Breakdown: Modeling Minsky's 'Financial Instability
Hypothesis.'." Journal of Post Keynesian Economics 17(4): 607-635.
Marx, K. (1867). Capital. Moscow, Progress Press.
Marx, K. (1894). Capital Volume III, International Publishers.
2
Proxied in this model by the wages share of output; this is explicitly replaced by the rate of inflation in the next
section.
Minsky, H. P. (1977). "The Financial Instability Hypothesis: An Interpretation of Keynes and an
Alternative to 'Standard' Theory." Nebraska Journal of Economics and Business 16(1): 5-16.
Minsky, H. P. (1982). Can "it" happen again? : essays on instability and finance. Armonk, N.Y.,
M.E. Sharpe.
Pomeau, Y. and P. Manneville (1980). "Intermittent transition to turbulence in dissipative
dynamical systems." Communications in Mathematical Physics 74: 189-197.