Complexity International  


ISSN 1320-0682

Source:   Received: 01/07/1998
Vol 6:   Copyright 1998   Accepted for publication: 15/10/1998

The Nonlinear Dynamics of Debt Deflation

Steve Keen
Department of Economics and Finance
The University of Western Sydney Macarthur,
Campbelltown, 2560

1    Introduction

After an era of moderate to high inflation  lasting almost 30 years, the USA has recently begun to record falling consumer and intermediate goods prices. Prices in Japan have been static to falling (except for the immediate impact of the Asia crisis) since 1994, while several European countries have near zero rates of inflation.

Contemporaneously, several major East and Southeast Asian nations have experienced debt-induced economic collapses, and their largely foreign currency denominated private sector debts were subsequently dramatically amplified by market-driven  currency devaluations. The ``Asian crisis''  will have international repercussions, both through a fall in their imports and the flow-on efffects of their attempts to export their way out of their difficulties. The deflationary  tendencies already evident in the West, and in particular the USA, could thus be amplified.

The prospect of a sustained period of falling prices thus appears likely in the West. While inflation is more likely that deflation in Asia due to the impact of the currency collapses, these devaluations will exacerbate the problems of excessive private sector debt. As has been most obvious in the case of Indonesia, the market value of a currency is largely determined by speculator expectations of the country's ability to service its foreign currency debts.

These developments have led many observers to worry that a debt-deflationary process may have commenced in Asia, which may be partially transmitted to the West via aggressive exporting and consequent import price deflation. Significant public figures are also voicing the concern that policy experience gained during an era of inflation may be inappropriate during one of deflation. As Alan Greenspan commented, ``deflation can be detrimental for reasons that go beyond those that are also associated with inflation''[4]. To have any inkling of what the economic future might have for us, we have to consider economic theories of deflation in the context of private debt.

2    Economic Theory

Compared to the wealth of economic argument about what can be done to control inflation, there is remarkably little economic theory devoted to deflation.(1) The main contributions were made by Fisher during the Great Depression [1], to some extent Keynes[9], and Minsky[11, 12, 13]. Minsky's ``Financial Instability Hypothesis''  can be regarded as distilling the essence of these contributions.(2)

This is that a pure market economy is characterised by a fundamental asymmetry which can cause the debt to output ratio to rise over time, to levels which can be unsustainable. This asymmetry is easily put: firms incur debt  to finance investment during booms, but have to repay that debt during slumps. Since the cyclical path of a capitalist economy is itself asymmetrical, this results in the level of debt ``ratcheting up'' during a sequence of trade cycles. Under fairly general conditions, this process can reach a point at which the accumulated debt overwhelms the debt-financing capacity of the economy, thus leading to a Depression.

Conversely, according to Minsky's hypothesis, a mixed market-state economy avoids this tendency towards complete collapse because the counter-cyclical behaviour of government spending -- rising during slumps and falling during booms -- counters the tendency of the private sector to accumulate excessive debt. Minsky argues that increased taxation during a boom attenuates investment, thus limiting the tendency to over supply productive capacity and consequently borrow excessively during a boom, and that increased government spending during a slump provides additional cash flow to businesses which stops their debt exploding. This argument is in direct opposition to the attitude of conventional macroeconomic analysis to government spending, which has long since replaced the Keynesian advocacy of counter-cyclical government behaviour with an effective demonising of government deficits.

I have previously modelled these theories in the absence of price dynamics[5, 6, 8]. In this paper I extend the model to incorporate the impact of a variable price level, and inflation-dependent rates of interest.

3    The Basic Model

The foundation of this model is Goodwin's model   of cyclical growth[3], which was itself based upon the Lotka-Volterra  predator-prey  model of species interaction on the one hand, and Marx's income-distribution/employment model of the trade cycle on the other. Over one century later, his arcane language notwithstanding, the best expression of this model is still that given by Marx: 

a rise in the price of labor resulting from accumulation of capital implies ...accumulation slackens in consequence of the rise in the price of labour, because the stimulus of gain is blunted. The rate of accumulation lessens; but with its lessening, the primary cause of that lessening vanishes, i.e. the disproportion between capital and exploitable labour power. The mechanism of the process of capitalist production removes the very obstacles that it temporarily creates. The price of labor falls again to a level corresponding with the needs of the self-expansion of capital, whether the level be below, the same as, or above the one which was normal before the rise of wages took place ...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. [10, pp580-581,]

Goodwin showed that this could be modelled as a predator-prey system in which workers share of output played the role of predator, and the rate of employment the role of prey:(3)


where tex2html_wrap_inline910 is the wage to output ratio tex2html_wrap_inline912 , tex2html_wrap_inline914 is a nonlinear relationship between the rate of change of wages w and the rate of employment (known as the ``Phillips curve'' ),(4) tex2html_wrap_inline918 the rate of employment or employment to population ratio tex2html_wrap_inline920 , tex2html_wrap_inline922 the rate of growth of labour productivity, tex2html_wrap_inline924 the rate of population growth, and v the capital to output ratio tex2html_wrap_inline926 .(5) As is well known, this model generates a stable limit cycle. The model also has an easy verbal explanation. The first equation says that workers' share of output will grow if their wage demands (which are based on the level of employment) exceed the rate of growth of labour productivity; the second that the level of employment will grow if the rate of economic growth exceeds the sum of population and productivity growth.

4    First extension: Investment and Debt

The first step in extending this model is to replace the linear assumption that capitalists invest all their profits ( tex2html_wrap_inline934 in the previous model is the profit to output ratio tex2html_wrap_inline936 ) with the more realistic assumption that investment is a nonlinear function k() of the rate of profit tex2html_wrap_inline940 , where tex2html_wrap_inline942 is profit net of interest payments.(6) This does not disturb the underlying nature of the model, which still results in a stable limit cycle, but it sets the scene for the introduction of a finance sector.

Finance is introduced into the model by assuming the existence of a banking sector which exists solely to finance capitalist investment. The rate of change of debt in this system is thus simply interest on outstanding debt, plus new investment, minus gross profits:


where tex2html_wrap_inline950 represents gross investment (in what follows, depreciation is introduced at the constant rate of tex2html_wrap_inline952 p.a.). This produces the following three-dimensional system:


where d is the debt to output ratio tex2html_wrap_inline954 and tex2html_wrap_inline942 is the net profit share of output:


As is well known, a three dimensional system introduces the possibility of chaotic behaviour, and this particular model follows the inverse tangent route to chaos  first identified by Pomeau and Manneville[14].

5    Perturbation Analysis

As is easily shown, with the functional form chosen for the ``Phillips curve'', the equilibrium  value of employment is: (7)


There is an equilibrium value for profit share:


which corresponds to a rate of profit of approximately 5.4% and, given the investment function, an investment share of output of 16.5%. The equilibrium value for the debt to output ratio is:


Since the profit share is a linear combination of tex2html_wrap_inline910 and d (equation (4)), this gives the curious result that at the equilibrium, workers' share of output and ``bankers' share'' are in direct opposition to each other, whereas ``capitalists' share'' is constant. The actual expression is:


This is, unremarkably, significantly different to standard economic models of income distribution, which argue that remuneration reflects relative factor productivity and which are not equipped to deal with a return to accumulated debt.

This equilibrium vector is locally stable but globally unstable, a significant echo of Fisher's intuition in 1933 that the market system has an equilibrium which ``though stable, is so delicately poised that, after departure from it beyond certain limits, instability ensues''[1, p339,]. 

Even at this basic level, the model contains some important insights into the role of debt in a market economy, and the impact of the rate of interest in a model in which, in contrast to the standard IS-LM model ,  debt is explicitly accounted for.

Conventional IS-LM analysis argues that an increase in the interest rate will reduce investment (which is portrayed as a monotonically decreasing function of the interest rate, in contrast to this model's argument that the rate of profit determines the level of investment) and thus growth; however any impact on the accumulation of debt is ignored. The final equation in (3) indicates that, when debt is explicitly accounted for, it is possible for debt to overwhelm the system, even though the equilibrium rate of profit significantly exceeds the rate of interest.

The approximate 5 year period of the cycles should also be noted: this is similar to those of the basic 2 dimensional Goodwin model.



Figure 1: Wages Share and Employment near Equilibrium


Figure 2: Debt to Output Ratio near Equilibrium

Figure 2 shows the time path of the debt to output ratio, which rises in a cyclical fashion initially, but then also tapers towards its equilibrium value.

The phase diagram in Figure 3 and the period interactions shown in Figure 4 give a clear picture of the dynamics in this 3 dimensional system.



Figure 3: Wages Share, Employment and Debt near Equilibrium - View image in VRML


Figure 4: Period Interactions of Wages Share, Employment and Debt near Equilibrium

The initial conditions of slightly higher than equilibrium debt, workers share of output and employment leads to a downturn, as investment stagnates due to the resulting low rate of profit. The excess of profit over investment leads to debt being reduced, but the downturn eventually leads to falling wage demands, and this leads to a boost in investment well before debt is fully repaid. Debt then rises with rising employment as investment boosts output, only to lead eventually to rising wage demands which cut into profits and once again cut off investment. The cycle then continues, with the system tapering towards a stable equilibrium debt to equity ratio, wages share and rate of employment.

Conversely, as Figures 5 to 7 indicate, at rates of interest which exceed the equilibrium  rate of growth, the equilibrium vector is unstable. From a conventional macroeconomic point of view, this system would appear to be stable right up until the final crisis, since conventional macroeconomics dismisses the issue of income distribution as a topic for microeconomic analysis, and ignores the role of debt. This simulation began with all variables .05 below their equilibrium values:



Figure 5: Wages share and employment far from Equilibrium


Figure 6: Debt to Output far from Equilibrium

However, the equilibrium analysis of this model would indicate cause for concern, since the secular trend towards decreasing workers' share of output would indicate that debt must still be rising, as is evidenced in Figure 6. Eventually, the level of accumulated debt becomes so high that repayments on outstanding debt eliminate all profit, leading to a collapse in output and hence a Depression:

The phase diagram of this simulation makes the bifurcation in system behaviour as the interest rate rises graphically apparent. What was previously a stable ``volcano'' shaped phase diagram becomes an unstable ``vortex'' in which debt overwhelms the other system variables:



Figure 7: Wages share, employment and debt far from Equilibrium - View image in VRML


Figure 8: Period Interactions of Wages Share, Employment and Debt far from Equilibrium

There is a superficially unremarkable explanation for this phenomenon: with a rate of interest higher than the rate of growth, it is not surprising that debt eventually smothers economic growth. However this is a nonlinear system, and with initial conditions at a further remove from the equilibrium, it is possible for the model to undergo a debt blowout even when the rate of interest is significantly lower than the equilibrium rate of growth. Figure 8 shows the behaviour of the model with a 3 per cent rate of interest when wages share is initially .1 below, employment .05 below, and debt .05 above the equilibrium vector of:


The behaviour of this model thus clearly supports the Fisher-Keynes-Minsky contention that a pure market economy is fundamentally unstable, in that it is prone to fall into a debt-induced Depression from which there is no escape, bar ``resetting the debt clock'' via wholesale bankruptcy and debt moratoria. The next extension similarly supports Minsky's claim that the government sector's behaviour provides a homeostatic balance which controls and possibly eliminates this tendency to Depression.

6    Second extension: A Government Sector

Minsky's contention that countercyclical behaviour by government stabilises the market by constraining its tendency to debt accumulation is explored by introducing government spending and taxation as functions respectively of the rate of employment and the profit share of output.(8) This extension requires new definitions for profit share and net profit share:


where tex2html_wrap_inline996 and tex2html_wrap_inline998 are as defined in the glossary. This extension results in the following five dimensional model of a mixed market-state economy:(9)


The behaviour of this model is consistent with Minsky's hypothesis. The most intriguing aspect, from a complex systems  point of view, is that the addition of a government sector transforms a system which was locally stable (about the equilibrium) but globally unstable, into a system which is locally unstable but globally stable. At least half the eigenvalues of the linearised version have positive real part for all values of r, yet rather than leading to breakdown, the model is constrained by a chaotic limit cycle, as the following simulations indicate.

The second intriguing feature of this model is the relationship between government debt and the interest rate. As with the previous model, the equilibrium wages share of output is a negative linear function of the interest rate, but in addition the level of government debt is a rectangular hyperbolic function of the interest rate (see Figure 9):




Figure 9: Bifurcation in the equilibrium government debt


Figure 10: Far from equilibrium dynamics at low interest

Thus if the prevailing (real) rate of interest is below the rate of growth of output, then with the equilibrium values for t and tex2html_wrap_inline952 given by the parameter values used in these simulations, the equilibrium value of government debt is negative. Equally, if the rate of interest exceeds the rate of growth, the equilibrium value is positive. While the actual values differ substantially from equilibrium values because of the system's far from equilibrium dynamics, this negative/positive bifurcation remains in any simulation. Figures 10-12 show the behaviour of the model with an interest rate of 3% and a .01 deviation of all system variables from the equilibrium vector.(10)



Figure 11: Far from equilibrium dynamics at low interest


Figure 12: Far from equilibrium dynamics at low interest - View image in VRML

The phase diagram in Figure 13 makes it clear that the dynamics are now governed by a chaotic limit cycle.

The model behaviour on the other side of the bifurcation point differs in one highly significant way: whereas government debt stabilised at a low rate of interest, at a high rate of interest government debt continues to grow cyclically but exponentially. rising government deficits have been a feature of post-WWII economies, especially since the adoption of a ``fight inflation first'' strategy in the mid-70s in an attempt to control the rate of inflation. The cornerstone of this policy was tight monetary policy -- which meant high real interest rates. Figures 14 to 17 demonstrate the behaviour of the model with an interest rate of 5% and a .01 deviation of all values from the equilibrium vector.

The apparent paradox in Figure 15 -- the coincidence of a positive overall government burden on the economy and yet a growing accumulated government deficit -- is explained by the impact of the high rate of interest on the current level of outstanding debt, and the already high level of debt implied by starting from the equilibrium position. However a different initial condition with low or negative initial government debt could easily result in a surplus being accumulated by the government (see [5]), as opposed to the deficit shown here.



Figure 13: Far from equilibrium dynamics at high interest


Figure 14: Far from equilibrium dynamics at high interest


Figure 15: Far from equilibrium dynamics at high interest - View image in VRML

7    Third Extension: Commodity Prices

Fisher argued that debt accumulation on its own would not be sufficient to cause a depression,  but instead would give rise to cycles. However the model above indicates the accumulation of debt alone can lead to a depression -- as the end product of a series of business cycles -- as the fundamental asymmetry that firms incur debt during booms but have to repay it during slumps asserts itself. Deflation  is thus not essential to the occurrence of a depression, but it would accelerate the process, and exacerbate its depth by its impact upon the rate of bankruptcy.  Similarly, Minsky's argument that capital goods prices are expectations-driven[13, p64, 80,] implies that pro-cyclical movements in capital goods will exacerbate the accumulation of debt, thus hastening the onset of a depression in a market economy.

These issues can be explored by revising the basic system of equations to include consumer prices ( tex2html_wrap_inline1025 ) and capital goods prices ( tex2html_wrap_inline1027 ). We start with an income shares equation in nominal (money) terms:


where wages can be decomposed into a real wage, a consumer price index, and the level of employment (L):


The wage change relation is now in money terms:


On the other hand, the relations between labor and output, and output and capital, must now be expressed in real terms:


The introduction of a capital goods price index affects the amount paid by firms for investment goods, but the change in physical productivity continues to depend on the real increment to capital. A distinction is thus required between nominal investment ( tex2html_wrap_inline1029 ) which affects bank balances, and real gross investment ( tex2html_wrap_inline1031 ) which affects the capital stock:


This results in the following system of equations:


Leaving aside the issue of functional forms for the rate of change of the price indeces, this set of equations confirms Fisher's and Minsky's insights concerning the impact of commodity price deflation and capital goods prices. As can be seen from the debt relation, a high rate of commodity price inflation reduces the real debt burden, as Minsky emphasises, while conversely price deflation will lead, as Fisher asserts, to an amplification of the real debt burden. The rate of debt accumulation also depends on the ratio of the capital goods price index to the consumer price index, and since the tex2html_wrap_inline1033 ratio will rise during a boom, this will accelerate the process of debt accumulation. The price system thus apparently increases the instability of the market economy.

To proceed, functional forms for the rate of change of the price indeces must be provided. This introduces one of the most vexing issues in economics, since despite the confidence of economists that they resolved the issue of price determination in the ``marginal revolution'', their theory of price setting has been under attack since its inception[15] and is clearly invalid in a dynamic setting.

The theory argues that price is set by the interaction of supply and demand, where the demand price falls as price rises and the supply price rises under the pressure of diminishing marginal returns. This generates a function for profit tex2html_wrap_inline944 as the gap between total revenue TR and total costs TC, whose maximum with respect to quantity occurs where ``marginal revenue'' equals ``marginal cost'':


Sraffa's  1926 critique was directed at one of the foundations of this latter argument, that any resource could be regarded as fixed in any realistic analysis of production in a modern economy. His critique is amplified when one introduces a realistic notion of time, as opposed to Marshall's  deceit that time could discretely be divided according to the variability of inputs. The neoclassical price-setting schema is clearly static: it tells how to maximise profit with respect to quantity (and thus determine price), but not how to maximise profit with respect to time. Clearly the latter objective is primary in a dynamic setting, and it can easily be determined via the chain rule:


This equation indicates that, regardless of the nature of tex2html_wrap_inline1037 , the neoclassical ``profit-maximisation'' price will result in a zero rate of growth of profits over time -- which is hardly the objective of any existing corporation. Far from being the obvious, simple but insightful rule which most economists believe it to be, their theory of profit maximisation (and hence price determination) is intellectually equivalent to the advice that the cheapest way to drive from point A to point tex2html_wrap_inline924 is to travel at zero kilometres per hour. An alternative pricing model must therefore be used. One candidate which is tractable at both the micro and macro level is the Kaleckian  proposition that prices are set by a markup on prime costs, where these in turn are the wages bill and depreciation:(11)


In a simple model with a fixed capital to output ratio, the depreciation component has no impact on the rate of change of prices, so that the rate of change of prices is entirely a function of the rate of change of wages and the wages share of output:


When this is substituted into the model, it results in the following 4 dimensional system:


One final modification is necessary before proceeding to simulations: given a price level, the rate of interest is no longer a real rate but a nominal one, and must therefore be allowed to vary with respect to the rate of inflation. This extension is not straightforward, since the nominal rate of interest has the crucial peculiarity that it must be positive, with a minimum rate set exogenously. The interest rate also responds to inflation in a lagged fashion.(12) These aspects of the interest rate are captured in two functions,(13) one specifying the lagged reaction of interest rates to the inflation rate, the other ensuring that the interest rate cannot be negative even when the rate of inflation is. While the real world relationships are bound to be more complex than these, they enable a first-pass at modelling the complex relationship between prices and interest rates. Equation (23) specifies the lagged relationship between the inflation component of the rate of interest and the rate of inflation:


where tex2html_wrap_inline1041 is the inflation-determined component of the rate of interest, and T is the time lag between changes to the inflation rate and changes to the rate of interest. Equation (24) specifies the nonlinear summation of the base and inflation-determined components of the rate of interest:


where tex2html_wrap_inline1045 is a curvature factor which also puts the actual rate above the base rate at zero inflation. A base rate of 3% and a value for tex2html_wrap_inline1045 of .000009 results in the following interest rate and inflation relationship:


Figure 16: Inflation Rate and Interest Rate Function


Figure 17: A Debt-Deflation

The full model is now:


This model is capable of demonstrating ``Fisher's Paradox'',  that deflation can mean that ``the more debtors pay, the more they owe''[1, p344,]. The process of deflation turns a low nominal rate of interest into a high real rate, and the depressing effect of debt repayment commitments on investment causes output to plummet, thus accelerating the blowout in the debt to output ratio. A true Debt-Deflation results:

There are several notable aspects to this extended model. Firstly, price dynamics almost completely subsume the income distribution dynamics of the previous models.(14) This result improves upon the realism of the basic Goodwin predator-prey cycle, since one well-known stylised fact is the relative stability of income shares, which display a secular trend but little cyclical behaviour. Secondly the range of behaviours that the model can demonstrate are dramatically extended over the basic two demonstrated by the non-price model. Other initial conditions can result in: bouts of cyclical employment and inflation behaviour before either a stable outcome of a debt-induced breakdown; sustained inflation with relatively constant income shares and restrained debt to output ratios; sustained deflation with a secular collapse in workers' share of output; and undoubtedly many more cases which will be uncovered by more systematic simulation explorations.

Thirdly, breakdown now begins at quite realistic values of the debt to output ratio, and the actual collapse can precede the blowout in debts to some extent because of the depressing effect on investment of high real rates of interest, caused by the process of deflation.

8    Conclusion

From a complex systems point of view, the addition of price dynamics to the basic Keen-Minsky model is a double-edged sword. On the one hand, it provides an additional source of potential long-term stability, with inflation countering the tendency towards the accumulation of debt. On the other, it can accelerate the process of collapse -- and possibly dramatically reduce the stable region around the system's equilibrium. A full answer to this question will have to await future research.

From an economic point of view, this model demonstrates many of the facets of Fisher's Debt-Deflationary ``creed''. Given that a debt-deflationary process is well under way in East and Southeast Asia -- exacerbated in some instances by severe exchange rate movements which this model is not as yet equipped to consider -- the model contains several important insights for economic management.

Firstly, contrary to conventional economic wisdom, a debt-deflation is a possibility: the events in Asia are not necessarily just the result of peculiar institutional arrangements of those countries.

Secondly, either inflation or government deficits may be necessary to overcome a debt-induced collapse -- though such relatively harmless means of escape from the abyss may be rendered ineffective in a world in which finance is international and exchange rates are market-driven.

Thirdly, as is now becoming obvious even to our most conservative politicians -- if not conservative economists -- finance can play a destabilising role in a capitalist economy. Deregulated finance is a recipe for crisis, not efficiency.


Milton Friedman is the economist most identified with the drive to control inflation. What is less well known is that his preferred rate of inflation was actually minus 4 to 5 per cent a year[2]. The paper in which this result was derived completely ignored the existence of debt.
See [13, especially Chs. 1,] and [5, 6, 7, 8] for lengthier expositions.
In that the linear term in the wages share of output equation is negative and that in the employment equation positive.
See the Glossary for the form of this and other nonlinear functions used in this paper.
I have extended the model to include variable capacity utilisation, so that the capital to output ratio is not a constant, and variable technical change, so that the rate of growth of labour productivity is not a constant[8]. Both these modifications makes the basic 2 sector model unstable, and increase the volatility of the final debt-deflation model.
The term tex2html_wrap_inline944 will be retained for gross profit or output minus wages throughout. The term tex2html_wrap_inline942 will signify gross profit minus all other outgoings, which at this stage means interest on outstanding debt. In the next section, tex2html_wrap_inline948 will signify gross profit minus interest payments, and taxation minus subsidies.
With the parameter values used in the following simulations, which were derived by a nonlinear regression of Phillips's original data against the rate of unemployment. The same regression is used later in the price level section.
The rate of profit is tex2html_wrap_inline984 , where v is a constant in this model. To simplify exposition, I used the profit share as the argument to the investment function rather than the rate of profit
The paper does not consider government debt, because at this level there is no consideration of the redeployment of net government spending into aggregate demand. The level of government debt can easily be incorporated into the model via the relation , and its level (as a proportion of output) is visible in these simulations in the gap between tex2html_wrap_inline952 and t.
Kaleckians also argue that price is set, at least to some extent, to finance planned investment, though there clearly are competitive limits to this process.
Lagged responses are also relevant in many other parts of this model, and will be introduced in further research.
I am grateful to Trond Andresen for assistance on these issues.
Indeed in this particular simulation price effects do completely subsume income distribution, with worker's share of output remaining constant throughout. This is however an artefact of the particular initial conditions chosen.


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