The paper below, a work-in-progress, is a differential version of
a 'toy spreadsheet model' I posted on PEN-L a couple of months
ago. Many thanks for all the positive responses to it (honest, I
didn't get any negative ones).
The differential version is a bit more difficult and needs some
math. I've tried to make it as straightforward as possible and if
anyone wants anything explained more I can do my best. Also if
anyone spots bugs in it I'd be grateful to know.
Most important is that this version is stable where the difference
(spreadsheet) version is not, which is what renewed my interest
in it.
I still don't think it is a plausible model for the cycle but I
think it makes some serious points; in particular, I think it
shows FROP has a decisive role to play in explaining cyclic
behaviour and can't be ignored any longer.
I can't post the charts in ASCII but I can send them UUencoded
or snail mail to anyone who wants.
I could show the cycle was stable and deduce its amplitude and
shape; as yet I haven't been able to calculate its periodicity
or formally prove its stability, though this is pretty damn
obvious. Any help would be welcome.
Thanks to John Ernst and Phyllis Attwater specially for a very
useful discussion on capital-output ratios.
AN ENDOGENOUS PROFIT RATE CYCLE
Alan Freeman, University of Greenwich
1. THE EQUATIONS
Consider the following system:
K' = I (1)
I' = aS (S/K-R) (2)
As we will show, this system exhibits stable periodic cycles in
both I and K except at the stable fixed point K=S/R. I oscillates
between equal negative and positive extremes, while K oscillates
asymetrically between a minimum lower than S/K but greater than
zero, and a maximum higher than S/K. This holds for any initial
values of I and any positive initial value of K and the
parameters S,R and a.
The variable K represents the money value of capital stock; the
variable I is the money value of the rate of investment; S, a
constant, is the money value of profits per period. R is a
'target' profit rate and a a parameter.
Equation (1) states that capital stock grows or shrinks at a rate
equal to the rate of investment.
Equation (2) states that if the average profit rate falls below
the target rate, entrepreneurs in aggregate reduce investment;
the greater the gap, the more rapid the reduction. Conversely if
the profit rate rises above this target rate, investment rises in
the same manner. The change in investment is here represented as
a proportion a(S/K R) of the mass of profit S.
Charts can be plotted to illustrate with initial values K0 = 200,
I0 = 10, and for S = 50, R = 0.05 and a = 0.02, using a numeric
package such as Phaser , MathCad or Mathematica, or supplied on
request.
The qualitative 'mechanism' is as follows; begin at a 'typical
point' when the actual profit rate is below the target (K too big)
but I is positive. K continues to rise, further decreasing the
profit rate. Since S/K is below R, investment is falling but
remains positive until a certain point determined by a, R and
the initial conditions. Once investment falls below zero
(disinvestment), K begins to fall so that S/K is now rising
but I' initially negative. When K falls to the critical point
S/R, I' becomes positive because the profit rate is now above the
target rate, but investment is still negative. Thus K undershoots
until it reaches a minimum, also determined by the parameters and
initial conditions, when I is positive again and capital stock
begins to rise. When K reaches S/R from below investment is now
positive but begins to fall, resuming the cycle.
2. THE BACKGROUND
It is known that endogenous business cycles can be generated or
simulated using second-order linear equation systems, the best-
known early such example being the Samuelson (1939) Multiplier-
Accelerator model.
It is a generally accepted criticism of such linear models that
only one value of the control parameter produces self-sustaining
stable cycles. The problem therefore remains of accounting for
the two most salient observed features of modern capitalism,
namely the persistence of business cycles and the lack of a
stable equilibrium.
A nonlinear accelerator as well as other subsequent models
developed by Goodwin and his co-workers brought home the fact
that persistent cycles arise only if there are nonlinear terms in
the equations of motion. But in most such models, the rate of
profit as a determinant of investment behaviour has been either
incidental or absent, and the adjustment process has focussed
either on the interaction between investment and consumption or
output, or on the interaction between employment and wage levels.
Neither neoclassical nor Marxist thinkers have, to my knowledge,
constructed formal models in which the rate of profit itself
exercises the predominant influence on cyclic behaviour,
notwithstanding the importance which the rate of profit assumes
in Marxist theory, and notwithstanding the significant empirical
evidence uncovered, by authors in both schools, of profit rate
variations during the course of the cycle. This includes, for
example, a major Bank of England study on the rate of profit and
the 1974 recession.
Marxist interest in cycles in the early part of the century
generally focussed, from Rosa Luxemburg onwards (see Day 1981),
on the instability of the proportions of reproduction per se
whilst neoclassical and Post-Keynesian studies focussed either on
pricing behaviour or on the relation between investment and
output or consumption. Cross-dual models such as those developed
by Semmler, Flaschel and their co-workers (see Wegberg 1990)
occupy an intermediate position inasmuch as the Marxist concern
with intersectoral proportions is maintained but the price system
plays a genuinely dynamic role.
I devised this equation system to test whether stable cycles
could be generated purely on the basis of changes in the rate of
profit. The system therefore abstracts from all variations of
investment behaviour which might be influenced or determined by
changes in the labour market, the price level or by variations in
quantities consumed or produced. For this reason S, the share of
the money surplus available either for investment or private
capitalist consumption, is held constant and only capital stock
and investment may vary; investment itself being affected only by
the rate of profit.
These simplifying features do not purport in any way to represent
a real economy. The system is not at all the only such possible
and I am not at all inclined to treat it as a 'correct' or
empirically valid model or even develop it into one. It is
clearly not realistic, for example, that any particular profit
rate such as R should be singled out as one of its stable
features. Moreover it is empirically and theoretically evident
that the period of the business cycle is closely tied to the
turnover time of fixed capital, a fact that finds no
representation in these equations.
Nevertheless as discussed in section 5, the assumptions adopted
do not rule out changes either in the real or money wage or in
the size of the net money surplus, nor technical change. What
they do is insulate movements in investment or capital stock from
such changes so as to isolate the impact of the average profit
rate. This exercise in thought is an indication that from a
theoretical point of view the influence of the rate of profit in
the business cycle must be seriously reexamined.
2. FORMAL PROPERTIES OF THE SYSTEM
We can transform equations (1) and (2) to a canonical form, which
makes it easier to study its fundamental properties, by using co-
ordinates x, y in which x, the transform of K, is 0 when K=S/R
and -1 when K = 0, y being similarly rescaled. The equations of
this transformation are
x = K(R/S) - R/S hence K= (x+1)S/R and K' = x'S/R
y = IR/S hence I = yS/R and I '= y'S/R
This leads to
x' = y
y' = aS{ S/(1+x)*(R/S)-R} = aSR{1/(1+x)-1}= -aSR x/(1+x)
Writing the constant B in place of aSR gives
x' = y (3)
y'= - B x/(1+x) (4)
This is a plane autonomous system, meaning that time plays no
explicit role. This makes it possible to derive the equation of
the orbit (the relation between x and y, or K and I) directly by
dividing x' by y' to give
dy/dx= - Bx/y(1+x) (5)
Separation of variables leads to
ydy = - B x/(1+x)dx = B{x/(1+x)-1}
whence integration yields
y^2 = B{log(1 + x) - x} + C (6)
where C is the constant of integration. This gives a set of
closed curves in the x-y plane, corresponding to periodic
oscillations of K and I.
Some insight can be gained by noting that if y is eliminated to
give the second-order differential equation
x" = -B x/(1+x) (7)
which may be compared with the classical linear oscillator (and
the linearisation of this system)
x" = -Bx (8)
in which Bx is a restoring force proportional to the distance x
from equilibrium. This oscillates indefinitely in a fixed orbit
unless either damped or forced. If the restoring force is not
proportional to distance x, the system may be compared either to
a stiffening spring (force increases 'faster' than Bx) or a
'softening' spring (force increases 'slower' than Bx).
The denominator 1/(1 + x) modifies this behaviour asymmetrically;
for positive x it lessens the 'restoring force' with distance
from x = 0, behaving like a softening spring whilst for negative
x it increases it, behaving like a stiffening spring. Hence the
phase of the cycle for which investment is decreasing (x < 0) is
longer than the phase for which investment is increasing (x > 0),
so that on the one hand the profit rate is above the target for a
longer part of the cycle than below it, but on the other the plot
of investment against time is a reverse sawtooth, the leading
edge in which investment is rising being sharper than the
trailing edge in which it falls.
The second result does not accord with the observed pattern of
the business cycle, but as stated earlier our intention is not to
mimic the cycle. It is to demonstrate that, since cyclic
behaviour can be emulated in a system in which all variation due
to fluctuations in price and quantity has been abstracted from,
it would be rash to conclude that these factors can be the only
ones at work. Since cycles can be emulated when the impact of
profit rate variations is the only remaining factor, this factor
must be taken seriously in further research.
The orbits of the system are given by the equation
H(x,y) = y^2 + Bx - Blog(1+x) = C (9)
where H, we note in passing, is the Hamiltonian of the system,
sometimes interpreted as its 'energy'. It is perhaps worth noting
that y^2 corresponds to the integral of investment over time, so
it might be treated as the 'pump' of the oscillating system.
4. SHAPE AND STABILITY
As remarked, the orbit is a closed curve in the x-y plane, so the
system will perform stable cycles which return it to the same
point (x0, y0) from which it started. Since at this point the
equations are identical to the starting point, its period must be
fixed, though this does not tell us what it is.
To study the orbit's shape a brief look at the curve
z = log(1 + x) - x (10)
is helpful. It has a singularity at x = -1, which according
to our choice of coordinates corresponds to the point K = 0.
It is important to establish if x can fall below -1, therefore,
as this would correspond to a negative capital stock.
The curve has a maximum at x = 0 (K = S/R), at which point z is
equal to zero. This immediately yields the extrema of y from
equation 6, namely y = +-sqrt(C). This also highlights the result
that there are no real orbits for C < 0.
We turn before establishing the extrema of x to the dependence of
the orbit on the initial conditions and the parameter a, which
between them determine the value of the constant B.
Substituting initial values x0, y0 leads to
C = y0^2 - B{log(1+ x0) - x0}
whence (6) becomes
y^2 - y0^2 = B{log(1 + x) - log(1 + x0) - (x - x0)} (11)
The extrema of x are given by writing dx/dy = 0, which from (5)
leads to y = 0 and hence
x - x0 - log(1 + x) + log(1 + x0) = y0^2
log (1 + x) - x = log(1 + x0) - x0 - y0^2 (12)
As noted, if x tends to -1 or oo, log(1 + x) - x tends to -oo,
so that (since the orbit is a closed curve) if the extrema exist
they must lie in the range -1< x0 < oo. They will not exist
if the righthand side of (12) is above the maximum of the
lefthand side, namely x = 0, which means that the orbit exists
and yields values of x between 1 and for any values of y0 and
hence any real initial value of I provided x0 is also between -1
and oo , that is, for any real positive initial value K0 and any
positive B.
5. ECONOMIC SIGNIFICANCE
It is commonplace that the test of a theory lies in the
prediction of observed reality. The current tendency to reduce
this to numerical accuracy alone, econometrics maybe serving as a
softening spring. I want to focus on a more exacting empirical
touchstone for theory; its ability to explain the observed facts
without missing anything out. This must include not just
quantitative but qualitative facts. A theory which numerically
predicts prices for six years to within five percent, and fails
to account for a devaluation, is not empirically valid, not
because of size of the numerical deviation which may be
shortlived, leading to good tests of significance but because
of the qualitative importance of devaluations.
In the evolution of thought, theories may conquer qualitative
peaks before ironing out quantitative valleys. It is the vantage
point from which their glacial power depends. But a theory which
does not scale peaks can never scour valleys. From this point of
view the superiority of Galilean astronomy lay not in the
accuracy of its detail, but the fact that it could explain a
decisive qualitative observation the moons of Jupiter which
could not be explained otherwise.
The great merit of Goodwin's contribution to economic theory
surely lies in his insistence that two decisive qualitative
features of capitalism remain to be explained by theory, namely,
the persistence and stability of cycles. These, along with the
inequality of people and nations, are perhaps the two most
persistently-observed phenomena of modern capitalism. Their
explanation is a Holy Grail of serious economic research
although, unlike the Grail, they intrude on mundane life.
If an endogenous cycle theory were to explain all qualitative
features of a modern economy, its superiority over all accounts
requiring external interference would be almost too obvious to be
stated. Less obvious, but worth stating, is that such a theory
would have to be based on the most persistent and profound
features of a market economy, since business cycles have existed
as long as the market economy. Any explanation which resorts to
an ephemeral or transient feature of capitalism is surely open to
question, since the rigorous enquirer must ask why cycles still
occur when the designated feature is absent.
Less obvious still is the fact that if any transient phenomenon
of capitalism is excluded a priori from such a theory, it would
be equally suspect. It would be like a theory which accounted for
the Jupiter and all the planets but ruled out comets.
Consider the accelerator equation
K* = kY
where K* is desired capital stock, Y is capital, and k the
capital-output ratio. This is the basis for Goodwin's (1951) non-
linear accelerator model, the dynamic process of adjustment
consisting in the way in which investment K' reacts to
differences between K* and K, the actual capital stock.
The customary interpretation of k is a technical feature of the
economy, representing the physical productivity of capital.
Indeed, this is why it is normally treated either as a constant
or a consequence of technical change.
There are a number of unsatisfactory features of this equation as
it stands which we do not repeat (Goodwin 1951 summarises many
of them). Most decisive for us is the following subtle point:
since monetary assets form part of the stock of capital K (and
from the investors' viewpoint they compete equally for attention)
and since K and Y are money quantities, the assumed technical
relation between them does not in fact exist. For, during the
actual course of a slump the first and probably main process is
not the scrapping of capital stock but the migration of new
investment capital from productive to unproductive investment;
essentially into monetary assets.
As far as investment behaviour is concerned, there is no
operational distinction between a money asset and a productive
asset. Indeed if the value of money is rising in some real sense,
it makes perfect rational sense to transfer income into monetary
or monetisable assets. Whatever its effect on output, this has as
such no tangible effect on the stock of capital.
The strength of Keynes' critique is that the 'adjustment' of
capital stock to output under such circumstances may and
empirically does take place through a downward adjustment of
output, not an upward adjustment of capital stock. But this also
revises the capital-output ratio downwards, as is to be expected
if assets are transferred from production but remain part of the
general stock of capital. A key link in the chain of the
accelerator-multiplier construction is thus broken; the supposed
technical link between output and capital.
The link between the profit rate and the stock of capital,
however, can be stated independently of such technical relations.
It is simply the ratio between total money profits and total
money capital. This must hold, regardless of the underlying
physical ratios; it is among the most basic facts of a money
economy. Equally persistent, and equally independent of all
technical relations, is the tendency of capital to seek the
highest possible return and it seems to me prima facie more
plausible to seek the determinants of investment behaviour (as
did both Keynes and Marx) in the relation between this investment
and the return it yields; or at least to include this relation in
the account.
It may be felt that this emphasis excludes the process of
technical change itself from the account by focussing exclusively
on the apparently superficial appearance of money prices. This
would misunderstand what the equation system allows us to say.
What occurs when, through technical change, a more cheaply-
produced product replaces a more expensive one? According to much
traditional theory, the current value of the existing stock of
this product counts in production at its new, replacement value.
A growing body of theoretical research (see Freeman and Carchedi)
challenges this view with the obvious point that the value of any
item of capital stock in the inventory of the investor is not
what it costs now to replace it, but its money price at the time
of purchase. Depreciation, from this point of view, subsumes a
windfall gain for the producer of the newer, cheaper product and
a loss for the owner of the older, more expensive stock. The most
extreme example of this I know of appeared in the August 1995 US
edition of PC Week, in which a 1990 Cray supercomputer costing
$12 million was advertised for $30,000 'or nearest offer'.
>From the point of view of society as a whole, therefore, the
money value of capital stock - the sum on which the profit rate
is calculated - is equal, not to the replacement cost of this
stock but to its historical money cost.
This being so, a full account of the dynamics of investment must
accept that the equation
K'=I
is, in money terms, not merely an approximation but an exact
truth.
Under this fundamental truth may lie a multiplicity of physical
realities. Disinvestment in money terms, for example, may result
either from physical scrapping or from simply replacing existing,
older and more expensive physical stock at the lower price made
possible by technical advance, the money surplus pace Luxemburg
being disposed of unproductively. In short, it is sufficient that
the process of technical change continue at a reduced rate,
without real expansion, for the slump phase of production
representable by the above model to apply.
Equally, the bald assertion of a constant money volume of profit
by no means precludes substantial variation in the real wage; S
is a summary variable which expresses the combined interaction of
changes in both productivity in the consumer goods sector and the
money wage. In conventional terms it simply expresses a
combination of costs which rises in total at the same rate as
productivity. What the assumption does is insulate the investment
mechanism from the details of this interaction, forcing our
attention onto the underlying interaction between the volume of
investment in money terms and the rate of profit in money terms.
Consequently the simplicity of equations dealing with money
capital stocks, and their relation to investment flows and profit
rates, belie the complexity of the physical relations they are
capable of expressing. In particular, if through value analysis
we can isolate the effects of changes in productivity from
ephemeral changes in the aggregate price level, we find that such
variables act, like energy in physics, as the basis for
establishing the constants of motion of an otherwise complex
system perhaps the nearest that economics can get to a global
expression of the ancient goal of an essentially simple summary
of its complex law of motion.
BIBLIOGRAPHY
With the exception of keynote articles the cited works are not
intended as a complete review of the literature but as a
selection of more recent surveys from which the reader may
identify works in the field.
Attfield, C.I.F., Demery, D. and Duck, N.W. (1992). Rational
Expectations in Macroeconomics. Oxford: Blackwell
Day, Richard B. (1981). The Crisis and the 'Crash': Soviet
Studies of the West 1917-1939. London:Verso
Dore, Mohammed H.I. (1993). The Macrodynamics of Business Cycles.
Oxford: Blackwell.
Drazin, P.G. (1992). Nonlinear Systems. Cambridge: CUP.
Freeman, A. and Carchedi, G. (1995) Marx and Non-Equilibrium
Economics. Aldershot:Edward Elgar.
Goodwin, R.M. (1951) 'The Non-linear Accelerator and the
Persistence of Business Cycles', Econometrica 19.
_______. (1992), Chaotic Economic Dynamics. Oxford: OUP
Goodwin, R.M., Krueger, M. and Vercelli, A. (eds) (1984) Non-
Linear Models of Fluctuating Growth. Berlin:Springer-Verlag.
Mullineux, A., Dickinson, D.G. and Peng, W. (1993) Business
Cycles. Oxford:Blackwell.
Samuelson, P. A. (1939). 'Interaction between multiplier analysis
and the principle of acceleration', Review of Economics and
Statistics, XXXI, pp75-78.
Wegberg, Marc van. (1990) 'Capital Mobility and Unequal Profit
Rates: a Classical Theory of Competition by Boundedly Rational
Firms', Review of Radical Political Econonomics Volume 22 Numbers
2 and 3.
