Cambridge Journal of Economics 2003, 27, 25–48
Capitalism’s growth imperative
Myron J. Gordon and Jeffrey S. Rosenthal*
A capitalist ﬁrm operating in a competitive market is subject to a growth imperative,
because uncertainty about the proﬁt rate under a no-growth policy makes the ﬁrm’s
prospects highly unattractive in ﬁnite time and bankruptcy practically certain in the
long run. A no-growth policy determines consumption and investment so that they
and capital would remain constant over time if the latter’s expected return were
realised with certainty. Simulation is used to arrive at the probability of bankruptcy
by the end of t periods and the expected values of capital and money, for relevant
combinations of time and uncertainty under successively more realistic models of a
no-growth ﬁrm in a competitive market. The sensitivity of the results to variation in
the parameters in each of the models is evaluated. Finally, we establish that a
plausible growth policy may achieve growth, but the problem of bankruptcy is not
resolved.
Key words: Growth, Bankruptcy, Capitalist system, Investment
JEL classiﬁcations: D8, D92, G32, G33, L1
Introduction
The primary purpose of this paper is to establish that a capitalist enterprise operating in a
competitive environment, be it a proprietorship or a corporation, is subject to a growth
imperative. By a growth imperative, we mean that the enterprise requires the expectation
of a positive growth rate, probably one that is well above the physically feasible rate of
growth for an actual closed competitive capitalist system. A positive and probably high
mean rate of growth is necessitated by the facts that the actual proﬁt rate is uncertain, and
its realisation varies over a very wide range. This high variability makes an enterprise with
a zero or negative mean expected value for its growth rate face a future in which
bankruptcy is practically certain in the long run, and has an intolerably high probability in
the short run, while providing little or no compensating beneﬁts by way of growth in
income or wealth until bankruptcy takes place. In fact, income and wealth can be
expected to fall over time. By a competitive capitalist enterprise, we mean one in which
the enterprise buys and sells in markets in which the government or other regulatory
authority plays no role in determining the prices and quantities for what it buys and sells.
Manuscript received 20 November 2000; ﬁnal version received 15 May 2001.
Address for correspondence: Professor M. J. Gordon, Faculty of Management, University of Toronto, 105 St
George Street, Toronto, Ontario M5S 3E6, Canada; email: gordon@rotman.utoronto.ca, and jeff@math.
toronto.edu.
* Professor of Finance, Rotman School of Management, and Professor of Statistics, respectively, both at
University of Toronto. We beneﬁted from comments on an earlier draft by John Cornwall, George Frankfurter,
Robert Solow and two referees of the Cambridge Journal of Economics.
© Cambridge Political Economy Society 2003
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26 M. J. Gordon and J. S. Rosenthal
The reliance on competitive markets for specialisation and exchange and for the determination
of each capitalist’s proﬁt is the source of the high variability in the proﬁt rate and
the necessity of a positive and probably high mean expected rate of growth.
It is widely if not universally accepted that growth is desirable. Keynes gave rise to a
macroeconomic theory under which investment is not only desirable for growth in output
in the long run, but necessary to avoid stagnation and unemployment in the short run.
We make the stronger claim here, that growth is necessary for tolerable prospects for
future survival for each capitalist and perhaps for the system as a whole. Some business
leaders make the same claim, but they argue that innovations in technology, marketing,
etc. by their competitors force them to participate in the quest for competitive advantage.
To our knowledge, no theory of a capitalist ﬁrm or individual has been advanced under
which a policy intended to maintain it at a stationary level is certain or even highly likely
to lead to its collapse in the long run and offer highly unattractive prospects in the short
run.
Neoclassical theory, mainstream economic theory, offers a radically different view of a
capitalist ﬁrm. Nothing is easier for it to do than remain in a stationary state, and for a real
person, growth is a matter of taste. It is no more than preference between present and
future consumption.
We say it is capitalism that is subject to a growth imperative, because the same is not
true of a pure socialist system. Regardless of what other problems are faced by a socialist
system, its own operation does not subject it to a growth imperative. It will be seen that,
notwithstanding a common technology of production, the difference between how the
ﬁrm is administered in the two systems gives rise to a growth imperative in one and its
absence in the other. The propositions that a pure socialist system offers security and
stagnation, while a pure (competitive) capitalist system offers insecurity and growth have
been recognised, but they have not been investigated rigorously. What we shall do here is
explain how a capitalist system generates insecurity and growth, and then consider what
capitalists do to deal with the insecurity.
The next section presents a simple model of a capitalist ﬁrm and establishes what
happens to its probability of survival and its wealth over time under a no-growth policy,
both conditional on its survival and without that condition. We ﬁnd that it is certain to go
bankrupt in the long run, and simulation of the model reveals that the ﬁrm does poorly in
ﬁnite time regardless of its proﬁtability and other relevant variables. Section 2 reviews the
treatment of growth, uncertainty and bankruptcy in the economics literature. Prior to
Keynes, the problem of uncertainty about the future was resolved by assuming that the
uncertain future value of a variable could be represented by its mean. Keynes’s dissatisfaction
with this solution to the investment problem motivated a post-World War II
literature on investment under uncertainty and risk aversion. The mainstream theoretical
literature established that the utopian properties of a perfectly competitive capitalist
system are preserved in the face of uncertainty and risk aversion. This was accomplished
and the feasibility of a stationary state was maintained, we shall see, by ignoring or
trivialising the problem of bankruptcy.
Section 3 introduces a number of reﬁnements in the model that make it more realistic.
Bankruptcy in the long run is not shown to be certain in every case, but its probability is
very high for combinations of uncertainty and time that are large. Finally, Section 4
establishes that investment and expenditure policies to achieve growth realise that objective,
but they do not increase the probability of survival. Far more is needed.
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Capitalism’s growth imperative 27
1. A simple model of capital
This section will present a model of a capitalist, i.e., a proprietor or a corporation, that is
simple, powerful, and, we believe, useful for understanding their behaviour on the micro
level and the system’s macro behaviour. The model incorporates physical and nominal
capital, uncertainty about the proﬁt rate on capital, consumption, investment in both
forms of capital, and bankruptcy, all in a plausible manner. There are also important
omissions, as we shall see, among them labour, government and monetary policy.
1.1 The model
The wealth of a capitalist, also called a ﬁrm in what follows, at the start of period t is
comprised of its capital valued at cost, K(t), and the nominal amount of its net monetary
assets or money, M(t). Their sum is the capitalist’s wealth,
W(t) ϭ K(t) ϩ M(t) (1)
The initial value of K(t), K(1), is given, and the change in K(t), from one period to the
next is given by
K(t ϩ 1) ϭ K(t)[1–␭] ϩ I(t) (2)
where ␭ is the rate at which capital depreciates in productivity from one period to the next,
and I(t) is the expenditure of money on additions to the gross stock of capital.
The ﬁrm’s quantity of net monetary assets is its cash, receivables and bonds, less its
payables and debt. M(1) is given and M(t ϩ 1) is raised above M(t) by M(t)r, the interest
on its initial balance and by P(t), the gross proﬁt on production. M(t ϩ 1) falls as a consequence
of the period’s investment and C(t), the consumption or dividend plus overhead
costs incurred by the ﬁrm. Hence,
M(t ϩ 1) ϭ M(t)[1 ϩ r] ϩ P(t)–C(t)–I(t) (3)
The actual gross proﬁt on production during t is
P(t) ϭ ␣(t)K(t) (4)
with ␣(t) assumed for simplicity to be a normally distributed random variable with mean
␮ and standard deviation ␴.
The capitalist’s consumption plus administration cost is the fraction ␩ of K(t) or
C(t) ϭ ␩K(t) (5)
and the gross investment in capital is
I(t) ϭ g␭K(t) (6)
Here, g is the ratio of the investment expenditure to the depreciation on the capital, so that
g ϭ 0 means that gross investment is zero and g ϭ 2 means that investment in t is twice the
depreciation on the existing stock.1
The economic security of a ﬁrm increases with the ratio M(t)/K(t), and insecurity rises
as M(t)/K(t) falls below zero. There is a lower limit to the ratio; that limit is no lower than
M/K ϭ –1, and when that limit is reached, the ﬁrm is bankrupt.2
The creditors then take
1
Capital here includes only ‘ﬁxed’ assets subject to depreciation. Inventory has different properties, and a
possible extension of the model (that will not be undertaken here) is to include both types of capital.
2
A widely used measure of ﬁnancial position is the debt–equity ratio, –M(t)/[K(t)+M(t)]. We could just as
well have used it as our bankruptcy condition.
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28 M. J. Gordon and J. S. Rosenthal
and liquidate the ﬁrm’s assets for whatever can be recovered of the money owed to them.
The corporation ceases to exist, and the proprietor ceases to be a capitalist. The upper
limit on –M/K is in fact less than one, owing to the loss on and costs of taking over and
liquidating the assets of a bankrupt ﬁrm. This condition for bankruptcy may seem to be
excessively harsh and, today, bankruptcy law is far kinder to ﬁrms in ﬁnancial distress, as
will be seen later. However, this seemingly harsh policy would be the practice in a
competitive capitalist system, where the law does not overrule private contracts.
1.2 The no-growth policy
A ﬁrm operating in competitive markets is subject to a growth imperative because of the
variability in its proﬁt rate. The gross proﬁt on production, that is, earnings before
deducting administration or overhead expenses, depreciation and interest, expressed as a
fraction of capital employed by the ﬁrm, can ﬂuctuate over a wide range from one period
to the next. This is captured by the mean and variance of ␣(t) in equation (4).
The basis for our claim that a capitalist enterprise operating in competitive markets is
subject to a growth imperative is that a no-growth policy makes bankruptcy practically
certain in the long run and has an intolerably high probability in the short run, while
providing little or no compensating beneﬁts. We deﬁne a no-growth policy as one under
which C and I are determined so that the values of K and M would remain constant over
time if the expected value of the proﬁt rate were realised with certainty. That policy is
realised under the model represented by the above equations when the expenditure rate
q ϭ g␭ ϩ ␩ ϭ ␮, and g ϭ 1 and r ϭ 0 (7)
The condition that g ϭ 1 guarantees that K remains constant over time by making
investment equal to depreciation. The condition that q ϭ ␮ makes the expenditure in each
period equal to the average proﬁt rate. Hence, with r ϭ 0, we have C, I and M as well as K
constant over time when the future is certain, that is, when ␣(t) ϭ ␮ for all t.
With the equations (7) true, the model satisﬁes a martingale property, so that the ﬁrm is
certain to go bankrupt in the long run. This is demonstrated in the Appendix (Theorems 3
and 4).1
An explanation of why bankruptcy is inevitable sooner or later that is an oversimpliﬁcation
may nonetheless help understand what is going on. Let bankruptcy take place
when M/K ϭ –1 or less and ignore for the moment the investment and consumption–
administration expenditures. Note that the operating proﬁt or loss for a period can be a
loss of any magnitude, such as 300% of K or more. If M/K at the period’s start is low
enough so that a loss of 300% or more makes M/K ϭ –1 or less, the ﬁrm is bankrupt in that
period. The likelihood of bankruptcy increases with the variance of the proﬁt rate and
inversely with M/K. Also, the likelihood it will be sooner rather than later is increased by
the size of the periodic investment and consumption–administration expenditures as a
fraction of K.
1.3 Simulation of model
The proof that a capitalist enterprise is certain to go bankrupt in the long run may remind
some readers of Keynes’s response to the proof that the long-run tendency of a perfectly
competitive capitalist system is full employment: ‘In the long run we are all dead.’ An
1
The appendix establishes that bankruptcy is certain in the long run under weaker assumptions than those
stated above. In the course of what follows, we shall develop more realistic models of the ﬁrm which will
recognise r > 0 among other things, and we shall consider their consequences for the no-growth policy and the
ﬁrm’s long-run survival.
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Capitalism’s growth imperative 29
actual capitalist is more concerned with the probability of bankruptcy over a reasonable,
ﬁnite time horizon.
To arrive at some idea of the consequences of a no-growth policy in ﬁnite time for a ﬁrm
described by the above model, we simulated its fortunes. We arrived at the probability of
bankruptcy by the end of t periods, and the expected values of K and M, both conditional
on survival for t periods and without that condition. For each simulation, we calculated
the probability of bankruptcy by the end of t periods for t ϭ 2, 10, 25, 50 and 100.
Bankruptcy was assumed to take place when the debt reaches –M(t) ϭ 0·5K(t).1
The
calculations were made for ␴ ϭ 0·0, 0·10, 0·20, 0·40 and 0·60. The simulation was carried
out for a system with 10,000 ﬁrms, each one initially identical and with subsequent
fortunes that differed only with respect to the random outcome with respect to ␣(t).
Table 1 presents the results of a simulation under which K(1) has the arbitrary value of
$1,000 and M(1) ϭ 0. Regardless of the initial K(1) value, all that matters is M(1) relative
to K(1). Reasonable values assigned to the relevant variables that satisfy the no-growth
condition were depreciation rate ␭ ϭ 0·10, consumption–administration rate ␩ ϭ 0·15,
mean gross proﬁt rate ␮ ϭ 0·25, and investment ratio g ϭ 1. The expenditure rate is
q ϭ g␭ ϩ ␩ and with r ϭ 0, the no-growth condition is satisﬁed.
We see in Table 1 that with ␴ ϭ 0 and the future certain, K ϭ 1,000 and M ϭ 0 in every
future period. Also, gross investment ␭K ϭ 100 and consumption ␩K ϭ 150 in every
period. With ␴ Ͼ 0, the investment policy keeps the mean K ϭ $1,000 as long as the ﬁrm
survives, but that is only true conditional upon the ﬁrm surviving for the t periods. Most
important, the probability of bankruptcy rises as ␴ rises above zero and as we go further
into the future. For example, with ␴ ϭ 0·40, the probability of bankruptcy rises above
one-half by t ϭ 10, and with ␴ ϭ 0·60, the probability of bankruptcy rises to 0·84 by
t ϭ 50.
With the mean values of K and M over time conditional on survival, M rises sharply
with ␴ as well as t, because the surviving ﬁrms are proﬁtable and the no-growth policy
limits them to putting the excess proﬁts in the bank. The unconditional expected value of
M also rises with both ␴ and time, but the unconditional values of wealth, that is W ϭ K ϩ M
fall for each value of ␴ Ͼ 0 as t rises. The important attributes of W are that it falls as t rises
for each value of ␴ and it never rises above its initial value of W(1) ϭ 1,000. Note how
sharply the unconditional expected value of K falls with time.
We simulated the model represented by equations (1)–(6) for a number of other
combinations of the parameters that satisfy the no-growth conditions of equation (7). We
were surprised to ﬁnd that all of these produced exactly the same output that we see in
Table 1.2
For instance, ␭ ϭ ␩ ϭ ␮ ϭ 0 also satisfy the no-growth condition, and that
simulation also results in exactly the same survival probabilities and K and M values as in
Table 1. We then found that to be no coincidence: Theorem 1 in the Appendix proves
that it must take place.
1
It may be wondered why bankruptcy takes place when –M(t) reaches 0·5K(t) instead of K(t). When –M(t)
reaches 0·5K(t), the debt covenant allows the creditors to call their loans. The market value of the ﬁrm’s
capital will then be below K, its book value, so that the ﬁrm is unable to meet their demand or reﬁnance the
debt elsewhere, and it is wiped out. Making bankruptcy take place when—M(t) is closer to K(t) would not
result in any substantive change in the conclusions reached. The critical assumption is that government does
not intervene in credit markets to govern or overrule the terms of debt contracts in the interest of debtors.
2
This takes place when the random number generator that determines the sequence of proﬁt rates for each
ﬁrm starts at the same place as we go from one combination of the parameters to another. Starting a
simulation with a new random number produces different numerical results that are not statistically
signiﬁcant.
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30 M. J. Gordon and J. S. Rosenthal
Table 1. Survival probabilities and expected values of capital t periods in future for a capitalist ﬁrm that
follows a no-growth policy. Expenditures depend solely on capital, as in equations (5) and (6) with parameter
values: ␭ ϭ 0·10, g ϭ 1·0, ␩ ϭ 0·15, ␮ ϭ 0·25, ivr ϭ 0·10, q ϭ 0·25
t Fraction bankrupt Conditional on survival Unconditional
K M K M
Uncertainty ␴ ϭ 0·00
2 0·0000 1,000 0 1,000 0
5 0·0000 1,000 0 1,000 0
10 0·0000 1,000 0 1,000 0
25 0·0000 1,000 0 1,000 0
50 0·0000 1,000 0 1,000 0
100 0·0000 1,000 0 1,000 0
Uncertainty ␴ ϭ 0·10
2 0·0000 1,000 1 1,000 1
5 0·0054 1,000 3 995 3
10 0·0663 1,000 42 934 39
25 0·2504 1,000 196 750 147
50 0·4230 1,000 422 577 243
100 0·5669 1,000 752 433 326
Uncertainty ␴ ϭ 0·20
2 0·0069 1,000 7 993 7
5 0·1303 1,000 92 870 80
10 0·3086 1,000 284 691 196
25 0·5272 1,000 717 473 339
50 0·6572 1,000 1,207 343 414
100 0·7503 1,000 1,922 250 480
Uncertainty ␴ ϭ 0·40
2 0·1004 1,000 84 900 75
5 0·3735 1,000 427 627 268
10 0·5499 1,000 895 450 403
25 0·7084 1,000 1,823 292 532
50 0·7907 1,000 2,854 209 597
100 0·8487 1,000 4,373 151 662
Uncertainty ␴ ϭ 0·60
2 0·1929 1,000 210 807 170
5 0·4858 1,000 802 514 412
10 0·6425 1,000 1,551 358 554
25 0·7707 1,000 2,953 229 677
50 0·8358 1,000 4,522 164 742
100 0·8813 1,000 6,769 119 804
1.4 Uncertainty of the proﬁt rate
Table 1 makes clear that the probability of bankruptcy in ﬁnite time depends materially on
␴, the uncertainty of the proﬁt rate on capital. It would therefore be most desirable to have
some idea as to the plausible range of ␴. We have practically no direct data on that
statistic, but the price of a company’s stock moves more or less with its earnings, and a
statistic on which considerable data exists is the standard deviation of the return on the
US stock market in the past. The capitalisation weighted real return on all stocks traded
on the NYSE over the years 1926–90, had an arithmetic mean of 8·6%, a geometric mean
of 6·4%, and a standard deviation of 21·1% (see Siegel, 1992, p. 31). Individual stocks
had mean returns that were larger or smaller than 8·6%, and their standard deviations
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Capitalism’s growth imperative 31
were on average larger, since the return on the stock market as a whole beneﬁted from
diversiﬁcation. Also, ﬁrms traded on the NYSE did not operate in competitive markets as
described earlier. They all enjoyed varying degrees of monopoly power, acquired in order
to raise gross return and reduce risk. Consequently, we may speculate that ␴ ϭ 0·20 is
very low for an average ﬁrm that buys and sells in competitive markets. They may well
have returns with a standard deviation as high as or higher than 0·60, and 0·40 is a quite
conservative ﬁgure, in our judgement.
A further consideration in reaching a judgement on our value of ␴ is that the upper limit
on the loss on a share of stock over some time period is 100%,1
but there is no upper limit
on a capitalist loss expressed as a fraction of the capital employed. To see why, recall that
the rate of proﬁt (or loss) on a share of stock is the dividend plus the rise (fall) in price
divided by the initial price. Since the dividend and price cannot fall below zero, the largest
possible loss is 100%. The rate of proﬁt (loss) for an enterprise is the operating proﬁt
(loss) divided by the capital employed valued at its cost. Subject to the ability of the
enterprise to draw on a cash reserve or borrow from a bank, the loss in any period can be
a few hundred per cent. One of the advantages attributed to private enterprise by
comparison with government enterprise is that there is a limit on the ability of the former
‘to throw good money after bad money’ or get others to join in doing so.
1.5 Consequences
The consequences of a no-growth policy for a capitalist described by equations (1)–(6) are
quite unattractive. As stated earlier, bankruptcy is certain in the long run. In ﬁnite time,
consumption and the stock of capital conditional on survival remain constant: only the
stock of money grows, but its sole purpose is to defer bankruptcy, and it is not very
effective for that purpose. The unconditional expected values of total wealth decline over time.
The absence of growth in consumption and capital are an expected consequence of the
no-growth policy, but there is no compensatory beneﬁt in a high probability of survival in
ﬁnite time as well as the long run. Recall for instance that with ␴ ϭ 0·40, the probability of
bankruptcy rises above one-half by t ϭ 10. These dismal consequences for a no-growth
policy represented by equations (1)–(6) are true regardless of the values the capitalist
assigns to or enjoys for the investment, consumption, proﬁt and depreciation rates as long
as they satisfy the no-growth conditions of equation (7).
The ﬁrm referred to in the presentation of the above model is a capitalist ﬁrm for good
reason. The fundamental distinction between a capitalist and socialist ﬁrm is with regard
to its control, which is exercised most simply and effectively through ownership. In a
capitalist economy, the control is private, so that each ﬁrm has an owner who enjoys or
suffers the ﬁrm’s proﬁt or loss. That is the objective of control. The proﬁt, wealth etc. of
each capitalist ﬂuctuated more or less from one period to the next due to the vagaries of
operating in a competitive market. In aggregate over the large number of capitalists, the
1
This property of a stock has generated much interest in the comparative features of the arithmetic and
geometric mean rates of return on a security. The arithmetic mean is an average of N simultaneous returns,
while the geometric mean is the average rate at which wealth will grow over N periods. It is the annual return
that one can expect to earn in the long run – as time goes to inﬁnity. The geometric mean is below the
arithmetic mean to a degree that depends on the variance of the return, and if there is a non-zero probability of
a return of –100%, a portfolio that consists solely of this security will go to zero in the long run. This property
of the geometric mean return has generated interest in Kelly strategies, that is, portfolios that maximise the
geometric mean and combining them with a risk-free asset to limit the probability of bankruptcy (see
Bernouilli, 1954; Latané and Tuttle, 1967; Markowitz, 1976; MacLean and Ziemba, 1999). There will be
more on bankruptcy in the next section.
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32 M. J. Gordon and J. S. Rosenthal
aggregate values of K, M and the other variables are practically constant from one period
to the next. In a socialist economy, the manager of each ﬁrm does not enjoy the ﬁrm’s proﬁt.
The proﬁts of all ﬁrms ﬂow to the state, and the constant aggregate is distributed in some
way by the central authority. The insecurity and bankruptcy in our simple competitive
capitalist system is completely absent in pure simple socialist system. Its hallmark is
security. Of course, this socialist system might well have real-world problems of motivation
and growth. We shall not deal with that subject here beyond noting that actual
socialist systems that were fairly close to a pure socialist system, such as Stalin’s Soviet
Union and Mao’s China, were forced to take extraordinary measures to deal with the
problems of motivation and growth. Mao tried the Great Leap Forward and the Cultural
Revolution.
2. Growth, uncertainty and bankruptcy
Before going on to extensions of our model that make it a more realistic representation of
a capitalist and then considering the consequences of pursuing growth, we shall attempt
to review its relation to the relevant literature. Our objective is to establish where our
model departs from or advances existing theory. That takes place for the most part with
respect to growth, uncertainty and bankruptcy, and these topics, bankruptcy in particular,
will be emphasised in the review. The treatment of bankruptcy in mainstream (neoclassical)
theory has much in common with the treatment of death in religion. We shall see
that bankruptcy is ignored, denied, gloriﬁed or passed over quickly in the neoclassical
theory of the ﬁrm or individual.
2.1 Smith and Marx on growth
As indicated by the title of his great work, Adam Smith’s objective was maximising the
wealth of nations. To realise that objective, he argued that: (1) there should be specialisation
and exchange in markets free of government and other restrictions on free
competition among capitalists, and (2) each capitalist should maximise the share of
operating proﬁt devoted to the growth of capital. Operating proﬁt is value added less the
wages of production workers, and to the extent possible, it should be devoted to the
further accumulation of capital. Operating proﬁt devoted to consumption, management,
the arts, general welfare etc. should be minimised regardless of how useful such expenditures
might be (Smith, 1822, vol. 2, bk. 2, ch. 3). By implication, Smith disapproved of
feudal lords who devoted their entire surplus to their pleasures, war, the church and other
non-productive purposes.
Marx argued that the entire surplus was expropriated from the working class, and he
attacked that expropriation, while occasionally expressing admiration for the progress
made possible by its use. To our knowledge, Marx was the ﬁrst person to state that
capitalists were subject to a growth imperative. He wrote:
Moreover, the development of capitalist production makes it constantly necessary to keep increasing
the amount of the capital laid out in a given industrial undertaking, and competition makes the
immanent laws of capitalist production to be felt by each individual capitalist, as external coercive
laws. It compels him to keep constantly extending his capital, in order to preserve it, but extend it he
cannot, except by means of progressive accumulation. (Marx, 1906, p. 649)
For the most part, Marx attributed the behaviour of capitalists to the nature or personality
of the people who become capitalists. It was not made clear what, if anything, in their
circumstances drives them to subordinate all else to the further accumulation of wealth.
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Capitalism’s growth imperative 33
Luxemburg (1968) is the Marxist whose main thesis is that capitalism is subject to a
growth imperative. She relied on history to prove that capitalism has always relied on
predatory relations with non-capitalist systems at home and abroad to survive. She wrote
‘real life has never known a self-sufﬁcient capitalist society under the exclusive domination
of the capitalist mode of production’ (p. 348). Her effort at a more rigorous proof
of her thesis was ﬂawed, and that resulted in its rejection by other Marxists (see Sweezy,
1942, pp. 202–7). The most favourable treatment of Luxemburg’s thesis was by Joan
Robinson in an introduction to the edition cited (see also Gordon, 1987).
2.2 Neoclassical and Keynesian theory
Prior to World War II, the theory of investment was based on the assumption that the
future payoffs on an investment are known with certainty, or what commonly amounts to
the same thing, their uncertain future values can be represented by their expected values.
It could then be reasoned that the capitalist maximised the market value of wealth with the
investment or disinvestment that equates the marginal rate of return on investment with
the interest rate. The utopian properties of this investment decision were explored at great
length in the development of the theory of interest and capital. No reason apart from taste
and technology was found why a ﬁrm could not remain in a stationary state, and the same
was true of the allocation of income between current and future consumption on the part
of real persons.
The combination of the Great Depression, Keynes’s reputation and his literary skills
ﬁnally made underconsumption and underinvestment theories of aggregate demand
respectable. Kalecki (1954) developed a similar theory. Previously, the instability of actual
capitalist systems was attributed mainly to monetary policy, labour or ‘political’ events.
Keynesian economics made it acceptable to use ﬁscal as well as monetary policy to
manage the economy. From a theoretical viewpoint, the Keynesian consumption function
made private investment the critical economic variable, and Keynes railed against the
inadequacy of the prevailing theory of investment, both its failure to deal with uncertainty
about the future and the desire for security (see Keynes, 1936, bk. IV, esp. ch. 12).
Modigliani and Miller (1958) established the conditions under which the neoclassical
theory of valuation and investment under certainty holds in the presence of uncertainty
and risk aversion. To achieve this, they implicitly assumed that the corporation is nothing
more or less than a collection of separable assets and investment opportunities. That
makes it possible to value each asset or opportunity at a discount rate that is greater than
the interest rate by an amount that depends solely on its risk. Consequently, the value of a
ﬁrm is completely independent (apart from taxes and other market imperfections) of its
capital structure, its dividend policy and even its investment decision, the last because the
value of each investment opportunity may be captured as easily by selling it as by
undertaking it. Gordon (1994, ch. 2, 4, 5 and 6) summarised the widely recognised failure
of the theory to explain corporate practice. The theory was equally useless in explaining
how stocks are valued and in estimating the all-important cost of equity capital (see
Brigham and Gordon, 1968; Malkiel and Cragg, 1970; Fazzari et al., 1988; Gordon and
Gordon, 1997).1
1
Gordon and Shapiro (1956) and Gordon (1962) developed an alternative theory called the dividend
growth model, under which the value of a ﬁrm is the present value of its dividend expectation, and its cost of
capital is the dividend yield plus the expected growth rate in the value of its shares. The theory is widely used
in the practice of ﬁnance, and it is widely taught in business school ﬁnance courses. It has been banished,
however, from the theoretical literature on ﬁnance, probably because it recognises that growth is risky.
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34 M. J. Gordon and J. S. Rosenthal
Markowitz (1959) arrived at the set of portfolios that are mean-variance efﬁcient in one
period returns, and Sharpe (1964) made the further assumptions needed to arrive at the
Capital Asset Pricing Model (CAPM) for shares. The separability of the shares in a
portfolio made it easy for Hamada (1969) to relate the CAPM to the Modigliani–Miller
theory of corporate ﬁnance. However, for the CAPM to be true, the efﬁcient market hypothesis
must also be true: the expected return on a share must be equal to the average
realised returns. Unfortunately, 25 years of testing led Fama and French (1992),
Lakonishok and Shapiro (1986) and others to the conclusion that there is no evidence
to support the joint hypothesis that CAPM and EMH are true (see Frankfurter and
McGoun, 1999).
Papers by Samuelson (1969) and Merton (1969, 1971) established the consumption
and portfolio policy (allocation of wealth between a risk-free asset and a diversiﬁed
portfolio of risky shares) over time of an individual whose objective is the maximisation of
expected utility of future consumption. The individual is assumed to have constant relative
risk aversion and no minimum consumption, so the allocation is independent of the level
of wealth. The fraction of net worth invested in the risky share portfolio is found to be
π ϭ (␮ – r)/␤␴2
(8)
with ␮ the mean rate of return on the portfolio, r the risk-free interest rate, ␤ the investor’s
degree of relative risk aversion and ␴2
the variance of the portfolio’s rate of return. On
average, over all investors, π should be equal to one in a closed system without government
and π should be below one if investors own but do not owe the public debt. The data
are consistent with this conclusion, since ␮–r is about 0·06, and ␴2
is about 0·045. An
investor who puts all wealth in the risky portfolio has π ϭ 1, and that implies ␤ ϭ 1·33,
which is a reasonable degree of risk aversion. As ␤ and ␴2
rise above these values or the risk
premium ␮ – r falls below 0·06, π falls below one.1
2.3 Bankruptcy
There was no explicit recognition of bankruptcy in the Samuelson–Merton model. That
in effect made it an inﬁnitely terrible event that was never allowed to take place. Gordon and
Sethi (1997, 1998), recognised that there is life after bankruptcy by imposing a minimum
level on consumption. This extension of the Samuelson–Merton model produces intuitively
attractive results. Notwithstanding the assumption of constant relative risk aversion,
consumption expressed as a fraction of wealth and the fraction of wealth invested in the
risky asset both decline as wealth increases. As wealth falls toward zero, the individual
adopts a go-for-broke investment policy. They invest to the limit allowed by creditors,
since a conservative ﬁnancial policy makes bankruptcy certain within a few periods
1
Robert Lucas and his followers have developed a macroeconomic model in which the risk of holding the
market portfolio is the variance in the growth rate of per capita consumption, and that has given rise to the ‘equity
premium puzzle’ (see Mehra and Prescott, 1985; Kocherlakata, 1996). The latter reports on p. 47 that the
variance in the growth rate in aggregate consumption per capita is about 0·0013. That ﬁgure, combined with
an equity premium of about 0·06 would require a degree of relative risk aversion that is well beyond all reason,
if we are to have π=1 in equation (8). We do not understand the measure of risk used in the Lucas model,
unless the model is intended to represent no more than a pure exchange economy with the sole asset an
endowment of grain that is consumed, stored or loaned, and with the mean return on storage positive. In an
economy that employs land to produce grain, land would be the dominant asset; its value would be the
discounted value of its expected future output, and the variance in its return would be the change in the
discounted value of that expectation from one period to the next. Its risk would be far greater than the
variance in the growth rate in grain consumption. The Lucas model is so lacking in empirical relevance as to
make one wonder what makes its empirical results a puzzle.
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Capitalism’s growth imperative 35
anyway. The fraction of wealth consumed also rises as wealth falls toward zero, for the
simple reason that consumption will certainly not be allowed to fall below what it would
be in the event that bankruptcy takes place.
The theoretical literature that recognises the existence of bankruptcy also has treated it
in a curious way. Two theories of bankruptcy exist: the absolute priority rule and the
relative priority rule. Under the former, bankruptcy takes place when the value of the ﬁrm
falls to the amount of its debt, in which case the corporation is reorganised with all
creditors paid in full and the equity holders left with nothing. Neoclassical economists
advocate the absolute priority rule, claiming that a bankruptcy would then be a relatively
costless and unimportant event, like the reﬁnancing of an outstanding debt issue by a
strong corporation (see, for example, Senbet and Seward, 1995).
In fact, the question of bankruptcy does not arise with a determination that the market
value of a corporation’s assets has fallen below its outstanding debt. The question arises
when a corporation violates one of its debt contracts and the creditor demands payment.
What the law provides for in that event is the basis for the historical development of the
relative priority rule. In order to ensure an orderly disposition of the ﬁrm’s assets, the
corporation may and usually does obtain the protection of the courts when a creditor
demands payment. The court then may and usually does arrange a reorganisation of the
corporation in which the claims of the investors in the corporation, including stockholders,
are recognised on the basis of their priority, the weakest claims being scaled down
the most. Practically all business bankruptcies take place under the relative priority
rule, because the law makes it available to corporations. The costs of bankruptcy can be
substantial, according to neoclassical economists, only because the misguided intervention
of government has resulted in litigation, uncertainty and other costs.1
It may be wondered why we use the book instead of the market value of capital in
deciding when M/K puts the ﬁrm in bankruptcy. The use of market value instead of
book value for K, the real assets of a corporation, is widely recommended in the ﬁnance
literature. The practice is to use book value, because the alternative is more easily
advocated than carried out, for a number of reasons that will not be elaborated here.
Perhaps more importantly, M/K at market is far more volatile than a cost-based ratio, so
that it is quite impossible for a ﬁrm to maintain some debt–capital ratio at market value,
and fairly possible to do so at book value.
It may be argued that bankruptcy is of no concern on the level of an industry or an entire
economy, because we have birth as well as death: new ﬁrms replace old ﬁrms and life
goes on. However, the knowledge that they will be replaced rarely if ever makes death or
bankruptcy of no concern for real and corporate persons. What they do to avoid or delay
that possibility is what makes death or bankruptcy important for understanding the
behaviour of both real and corporate persons.
3. Extensions of the model
The model represented by the simulation in Table 1 is unrealistic in a number of ways.
For instance, the closed competitive capitalist system it is intended to represent has existed
1
For the development of this reasoning, see Senbet and Seward (1995), Baird (1996) and numerous other
essays in Bhandari and Weiss (1996). We could not ﬁnd a defence of the relative priority rule other than
Warren (1996), who argued that its widespread use makes it deserving of consideration. One of the reasons
why corporations elect to declare bankruptcy is that the relative priority rule eliminates the obligations to
workers created by union contracts.
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36 M. J. Gordon and J. S. Rosenthal
only in theory: its use may be justiﬁed only as a starting point for the consideration of more
realistic models. What we shall do now is explain or improve on what we consider to be
the most glaring or objectionable of the assumptions incorporated in the model.
3.1 Dependence on proﬁtability
Perhaps the most obvious objection to the plausibility of the model represented in Table 1 is
the assumption that consumption and investment are independent of income or proﬁtability.
They both move only with K, and it changes little from one period to the next.
In fact, a policy of making investment equal to depreciation makes K remain constant
at $1,000 over time for the surviving capitalists. Clearly, consumption and investment
should in part respond to income or proﬁtability, and changing our model to realise that
objective is accomplished by replacing equation (5) with
C(t) ϭ ␩K(t) ϩ max[ySP(t) or zero] (5a)
and by replacing equation (6) with
I(t) ϭ g␭K(t) ϩ max[hSP(t) or zero] (6a)
SP(t), the smoothed proﬁt, is an exponential average of its previous value and current
proﬁt.
SP(t ϩ 1) ϭ ␥SP(t) ϩ (1–␥)P(t) (9)
and the initial value of SP is SP(1) ϭ ␮K(1).
Equation (5a) is a Keynesian consumption function. It makes the reasonable statement
that in each period there is a ‘minimum’ consumption and administrative expense. It is
represented by ␩K(t), so that the minimum changes slowly over time with capital valued
at cost. In addition, the expenditure varies with expected income, but C(t) cannot fall
below ␩K(t), even though SP(t) Ͻ 0 is possible. Equation (6a) states that with g Ͻ 1 there
is a minimum expenditure on asset replacement regardless of proﬁtability that is less than
␭K. The actual level of investment depends also on proﬁtability, but this component of
investment cannot be negative, so that gross investment cannot fall below g␭K. Finally,
the sensitivity of C, I and SP to current proﬁt increases as ␥ falls toward zero.
Is a no-growth policy violated by making C and I partially dependent on proﬁtability?
No. We now replace the conditions q ϭ g␭ ϩ ␩ ϭ ␮ and g ϭ 1 with the conditions that the
expenditure rate and the investment rate,
q ϭ g␭ ϩ ␩ ϩ ␮(y ϩ h) ϭ ␮ and ivr ϭ g␭ ϩ ␮h ϭ ␭ (10)
With these conditions satisﬁed, and with 0 ≤ ␥ ≤ 1, K and M, as well as the decision
variables, remain in a stationary state when ␮ is certain, which is our condition for a nogrowth
policy.
The Appendix proves that when r ϭ 0 and either the ‘max’ constraints are removed
from equations (5a) and (6a) (Theorem 5), or ␥ ϭ 1 (Theorem 6) or ␥ is near 1 (Theorem
7), the above model satisﬁes a martingale property, which means that the ﬁrm is certain to
go bankrupt in the long run. The proof is difﬁcult under the max conditions of equations
(5a), (6a) but simulations with and without the max conditions under a wide range of
circumstances revealed that for every combination of ␴ and t, the probability of
bankruptcy was greater with the max conditions than without them. We therefore can say
with great conﬁdence that bankruptcy is certain in the long run for the system represented
by the above model.
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Capitalism’s growth imperative 37
To see what happens in ﬁnite time, we simulated the model under a range of values for
the parameters that seemed of interest. Of particular interest is what happens to the probability
of bankruptcy as the dependence of C and I on proﬁtability is raised by moving
␥ from 1 towards 0 and by moving y ϩ h from 0 towards 1, the latter with compensating
reductions in g␭ and ␩ to satisfy the conditions that q ϭ ␮ and ␭ ϭ g␭ ϩ ␮h. Either when
␥ϭ 1 or when y ϭ h ϭ 0 is true, there is no dependence of C and I on proﬁtability, and we
are back with the model of Section 1 that produced the output in Table 1. When ␥ ϭ 0 and
y ϩ h ϭ 1 with ␩ ϭ g␭ ϭ 0, there is complete dependence on proﬁtability, since ␥ ϭ 0 makes
SP equal to current proﬁts, and the minimum levels of both C and I are zero. The simulations
revealed that the probabilities of bankruptcy in ﬁnite time are substantially unchanged.
Table 2 presents the simulation results when C and I are made dependent on both
capital and proﬁtability by setting ␥ ϭ 0·5, with g and ␩ made positive and h and y taking
the values needed to satisfy equations (10). We kept ␮ ϭ 0·25 and ␭ ϭ 0·10, and the
values of g, h, ␩ and y were set to arrive at a reasonable distribution between the ‘ﬁxed’ and
‘variable’ components of C and I, while keeping the expenditure rate q ϭ ␮ and the investment
rate equal to the depreciation rate. Comparing the statistics in Tables 1 and 2 reveals
that partial dependence of C and I on proﬁtability provides less attractive prospects than
no dependence.
We simulated the proﬁtability model over a wide range of values for the above parameters
without violating the conditions in equations (10). In particular, we tried ␮ above
and below 0·25, raised and lowered the relative importance of C and I, and raised or
lowered the ﬁxed and variable components of C and I. Unlike the model of Section 1,
where the probability of bankruptcy and other statistics were independent of these
parameter values, here the statistics change in one way or another. However, as long as
the no-growth conditions of equations (10) are satisﬁed and r ϭ 0, the probability of
bankruptcy for each combination of t and ␴ varied over a limited range from one simulation
to the next. There are seemingly small changes in the parameters that produce
ridiculously large or small changes in K or M over time. We shall not burden the text with
a discussion of how reasonable they are.
Our conclusion is that the extension of our no-growth policy to make expenditure
partially dependent on proﬁtability does not eliminate the unattractive features of the
policy. The opposite is true. Bankruptcy is still inevitable in the long run. Furthermore,
the probability of bankruptcy in ﬁnite time is commonly raised for each value of t, and the
unconditional expected value of the growth in wealth is still negative. The lack of growth
in a no-growth policy is still not compensated for by a high probability of survival.
3.2 Other reﬁnements
The assumption that the interest rate r ϭ 0 was not made solely for convenience, in that
for r Ͼ 0 the theorems proved in the Appendix either could not have been proven, or they
would require a far more complicated argument. However, we established by simulation
that recognising interest paid on loans has a negligible impact on the survival probabilities
and on the expected value of K and M over a wide range of the values for r. Of course, a
very large initial wealth, a very high rate of interest on money loaned, very low values for
C/K etc., might make it possible and attractive for a capitalist to become a rentier and live
forever on M. However, these conditions are exceptional, and need not concern us.1
1
In the real world, we have rentiers who live solely on interest income, but that is possible only through a
redistribution of income through government and business debt. Real income is produced by K and not by M.
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38 M. J. Gordon and J. S. Rosenthal
Table 2. Survival probabilities and expected values of capital t periods in future for a capitalist ﬁrm that
follows a no-growth policy. Expenditures depend on capital and proﬁts, as in equations (5a), (6a) and (9)
with parameter values: ␭ ϭ 0·10, g ϭ 0·50, h ϭ 0·20, ␩ ϭ 0·05, y ϭ 0·50, ␮ ϭ 0·30, ␥ ϭ 0·50, ivr ϭ 0·10,
q ϭ 0·30
t Fraction bankrupt Conditional on survival Unconditional
K M K M
Uncertainty ␴ ϭ 0·00
2 0·0000 1,000 0 1,000 0
5 0·0000 1,000 0 1,000 0
10 0·0000 1,000 0 1,000 0
25 0·0000 1,000 0 1,000 0
50 0·0000 1,000 0 1,000 0
100 0·0000 1,000 0 1,000 0
Uncertainty ␴ ϭ 0·10
2 0·0000 1,000 1 1,000 1
5 0·0004 1,000 0 1,000 0
10 0·0044 1,001 3 996 3
25 0·0550 1,008 24 953 23
50 0·1734 1,030 79 851 66
100 0·3392 1,073 193 709 128
Uncertainty ␴ ϭ 0·20
2 0·0069 1,000 7 993 7
5 0·0708 1,005 33 934 30
10 0·1917 1,024 89 828 72
25 0·4174 1,085 238 632 139
50 0·5857 1,172 451 485 187
100 0·7147 1,303 795 372 227
Uncertainty ␴ ϭ 0·40
2 0·1004 1,000 84 900 75
5 0·3510 1,039 239 675 155
10 0·5607 1,125 438 494 192
25 0·7696 1,345 918 310 211
50 0·8728 1,676 1,558 213 198
100 0·9408 2,430 3,036 144 180
Uncertainty ␴ ϭ 0·60
2 0·1929 1,000 210 807 170
5 0·5050 1,084 505 537 250
10 0·7078 1,251 877 365 256
25 0·8770 1,730 1,843 213 227
50 0·9493 2,705 3,501 137 178
100 0·9881 6,405 10,380 76 124
Another possible objection to the models represented by Tables 1 and 2 is the growth in
M/K over time, conditional on the ﬁrm’s survival for the t periods. M grows beyond all
reason in relation to K in Table 1, and the growth in M/K is still large in Table 2, where
excess proﬁts are absorbed in part by C and I. M ϭ 0 is a strong ﬁnancial position, and
having MϾK, as takes place in these simulations, provides exceptional security. It is quite
possible that any particular capitalist is so obsessed with the fear of bankruptcy that all
excess cash is held as money. There are two polar alternatives for capitalist policy. One is
to spend all excess M on additional consumption, and the other is to devote it to the
further accumulation of capital. Under both policies with regard to the disposition of
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Capitalism’s growth imperative 39
excess M, its expected value is kept within reason, but the probability of bankruptcy rises
more sharply with ␴ and t. Hence, putting excess cash into some combination of consumption
and capital does not improve the prospects for survival. Quite the contrary, they are
reduced materially, the only compensation being that, conditional on surviving for t years,
the intervening years are somewhat more satisfying.
3.3 Serial correlation and mean reversion
An interesting and reasonable departure from our assumption that the expected rate of
proﬁt on capital is ␮ regardless of its realised value is the assumption that realised values of
␣(t) convey information about future values of ␮(t). This dependence of the expected
return on capital on its realised values and a tendency for the expected return to revert
towards some long run value is captured by the expression:
␮(t ϩ 1) ϭ c(1)␮(t) ϩ c(2)␣(t) ϩ [1 – c(1) – c(2)]␮(1) (11)
with the three coefﬁcients all positive and their sum equal to one. Bankruptcy in the long
run cannot be proven under the serial correlation and mean reversion that we have in
equation (11) (see the Remark just before Theorem 6 in the Appendix).
What happens in ﬁnite time would seem to depend materially on the relative values of
the three coefﬁcients of equation (11), and we simulated the model described in Table 2
with a limited range of values for the coefﬁcients of equation (11). Table 3 presents
the simulation results with ␮(1) ϭ 0·25, c(1) ϭ 0·5 and c(2) ϭ 0·4, and with the other
parameters the same as in Table 2. Comparison of Tables 2 and 3 reveals that in Table 3
the probability of bankruptcy rises much more rapidly with t and ␴ except for combinations
of t and ␴ that are both very large. The explanation is that in Table 2 at time t Ͼ 0,
the variation in K and W are due solely to ␴. In Table 3, serial correlation makes ␮ vary
among ﬁrms over time, so that ␮ as well as ␴ contribute to the variation K and W as t
increases. When t and ␴ are both very large, the few ﬁrms that remain are very wealthy and
proﬁtable. The important conclusions are that serial correlation makes the probability of
bankruptcy rise more rapidly in the short run, and its continued rise in the long run would
seem to make it inevitable unless another policy is adopted.
We repeated the simulation in Table 3 with a few other combinations of the coefﬁcients
of equation (11). For c(1) ϭ 0·4 and c(2) ϭ 0·5, for c(1) ϭ 0·6 and c(2) ϭ 0·3, and for
c(1) ϭ 0·4 and c(2) ϭ 0·3, the rates of growth in the probability of bankruptcy with ␴ and t
were remarkably close to the values in Table 3. On the other hand, the rates of growth in K
and M were raised, particularly for high values of ␴ and t, as c(2) was raised relative to
c(1). The rates of growth in K and M were reduced materially as the mean reversion was
raised by reducing the sum of c(1) and c(2). We did not carry out simulations for radically
different relative values for the coefﬁcients of equation (11), or for other values of the
other parameters in Table 3.
4. Growth policies
What might a ﬁrm do to reduce materially the high probability of bankruptcy that it
faces under the no-growth policies examined above? We shall examine three alternative
policies, with each policy taking as its starting point the parameters under the no-growth
policies in Tables 2 and 3. The three alternatives are: (1) reduce the expenditure rate
below the proﬁt rate, while leaving the investment rate equal to the depreciation rate; (2)
raise the investment rate above the depreciation rate while leaving the expenditure rate
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40 M. J. Gordon and J. S. Rosenthal
Table 3. Survival probabilities and expected values of capital t periods in future for a capitalist ﬁrm that
follows a no-growth policy. Expenditures depend on capital and proﬁts, with proﬁts inﬂuenced by serial
correlation as in equations (5a), (6a), (9) and (11) with parameter values: ␭ ϭ 0·10, g ϭ 0·50, h ϭ 0·20,
␩ ϭ 0·05, y ϭ 0·50, ␮ ϭ 0·30, ␥ ϭ 0·50, ivr ϭ 0·10, q ϭ 0·30, c(1) ϭ 0·5, c(2) ϭ 0·4
t Fraction bankrupt Conditional on survival Unconditional
K M K M
Uncertainty ␴ ϭ 0·00
2 0·0000 1,000 0 1,000 0
5 0·0000 1,000 0 1,000 0
10 0·0000 1,000 0 1,000 0
25 0·0000 1,000 0 1,000 0
50 0·0000 1,000 0 1,000 0
100 0·0000 1,000 0 1,000 0
Uncertainty ␴ ϭ 0·10
2 0·0000 1,000 1 1,000 1
5 0·0145 1,001 9 987 9
10 0·1304 1,021 102 888 89
25 0·3668 1,166 516 738 327
50 0·5250 1,442 1,220 685 579
100 0·6583 1,971 2,600 674 888
Uncertainty ␴ ϭ 0·20
2 0·0069 1,000 7 993 7
5 0·1565 1,014 123 856 103
10 0·3552 1,100 461 710 297
25 0·5621 1,540 1,697 675 743
50 0·6909 2,533 4,273 783 1,321
100 0·8010 5,751 12,671 1,144 2,522
Uncertainty ␴ ϭ 0·40
2 0·1004 1,000 84 900 75
5 0·3827 1,064 511 657 315
10 0·5529 1,313 1,423 587 636
25 0·7151 2,805 5,957 799 1,697
50 0·8277 9,646 25,667 1,662 4,422
100 0·9311 102,775 297,828 7,081 20,520
Uncertainty ␴ ϭ 0·60
2 0·1929 1,000 210 807 170
5 0·4828 1,120 947 579 490
10 0·6342 1,568 2,659 574 973
25 0·7815 5,156 14,636 1,127 3,198
50 0·8874 40,349 130,603 4,543 14,706
100 0·9707 2,222,473 8,154,835 65,118 238,937
equal to the proﬁt rate; and (3) do both. The no-growth policies in Tables 2 and 3 are
identical in their dependence of expenditures on capital and proﬁts, and they differ only in
that there is serial correlation and mean reversion in Table 3. Their presence is more
realistic than their absence: so we shall conﬁne our comparison of the three alternatives
with the no-growth policy to the simulations in which there is serial correlation and mean
reversion.
We could not establish theoretically whether bankruptcy is inevitable in the long run
under these growth policies. But in the simulations with the levels of ␮ and ivr and with
their excess over q and ␭g reasonable, the rise in the probability of bankruptcy with ␴ and t
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Capitalism’s growth imperative 41
are rapid enough to suggest that it is certain sooner or later. When the levels of ␮ and ivr
and their excess over q and ␭g are both large, this conclusion is in some doubt.
Our ﬁrst alternative policy, which is the reduction in the expenditure rate below the
proﬁt rate, was achieved by making q ϭ 0·25 and y ϭ 1/3, while leaving ␮ ϭ 0·30 and
h ϭ 1/6 at their values in Table 3. As expected, the probability of bankruptcy was reduced
somewhat, and unconditional expected values of K and M were raised, but the latter only
slightly. In short, the increased probability of survival was achieved at the cost of very little
growth in capital.
Under the second alternative policy, we made ivr ϭ 0·15 and h ϭ 1/3, while leaving
q ϭ ␮ at 0·30. Here, the probability of bankruptcy rose somewhat more rapidly with t and
␴ than in Table 3, while the unconditional growth rate in K was much greater and quite
fantastic for high combinations of t and ␴, where serial correlation has eliminated all but
the most proﬁtable ﬁrms.
The simulation of both departures from the no-growth policy appear in Table 4, where
the parameter changes from Table 3 are stated. Comparison of the tables makes clear that
the survival and growth statistics realise the best of both of the previous departures from
a no-growth policy. Here again, the elimination of the unproﬁtable ﬁrms by serial correlation
results in fantastic growth rates for the few ﬁrms remaining at high ␴ and t.
Clearly, the growth policy represented in Table 4 offers far more attractive prospects
than the no-growth policies examined in the previous section. There are, however, two
problems with this growth policy. Any one capitalist can follow a policy of making the
expenditure rate less than the proﬁt rate, but that policy cannot be followed widely in a
closed capitalist system without government. In aggregate, we must have the expenditure
rate equal to the gross proﬁt rate, unless there is lending to, or investment in, other
sectors—government, workers or the foreign sector. The other problem with the policy is
that it is Keynesian economics: with the limited exception of loans to government to
ﬁnance military expenditures, loans as well as gifts to others beyond charity are not
popular with capitalists. Welfare state policies were adopted in response to the horrendous
economic and political failures of capitalism over the years 1929–45, and they met
with great success over the years 1950 to 1973.Cornwall and Cornwall (2001) called the
period ‘the golden age’ of capitalism, but, for various reasons, the period was followed by
stagﬂation and the gloriﬁcation of the market and corporate rule.
What do real ﬁrms actually do to avoid the dismal prospects offered by a no-growth
policy? Real ﬁrms are radically different from their representations in neoclassical theory,
where they engage only in production; they are completely represented by a production
function; and they passively accept domination by a perfectly competitive market. Real
ﬁrms engage in a wide range of non-production activities in the pursuit of monopoly power.
These non-production activities range from research and development, advertising and
marketing, to bribing government ofﬁcials and overthrowing governments if necessary. For
high technology corporations such as Microsoft and Merck, expenditures in the pursuit
of monopoly power are much greater than their expenditures for production. Nike
is reported to engage in everything but the production of its products. A simple and
powerful measure of the monopoly power real ﬁrms enjoy is Kalecki’s Degree of
CJE 27/1 pp. 025-048 REV 13/11/02 3:00 pm Page 41
42 M. J. Gordon and J. S. Rosenthal
Table 4. Survival probabilities and expected values of capital t periods in future for a capitalist ﬁrm that
follows a growth policy. Expenditures depend on capital and proﬁts, with proﬁts inﬂuenced by serial
correlation as in equations (5a), (6a), (9) and (11) with parameter values: ␭ ϭ 0·10, g ϭ 0·5, h ϭ 0·33,
␩ ϭ 0·05, y ϭ 1/6, ␮ ϭ 0·30, ␥ ϭ 0·5, ivr ϭ 0·15, q ϭ 0·25, c(1) ϭ 0·5, c(2) ϭ 0·4
t Fraction bankrupt Conditional on survival Unconditional
K M K M
Uncertainty ␴ ϭ 0·00
2 0·0000 1,050 50 1,050 50
5 0·0000 1,194 246 1,194 246
10 0·0000 1,469 646 1,469 646
25 0·0000 2,731 2,490 2,731 2,490
50 0·0000 7,677 9,717 7,677 9,717
100 0·0000 60,680 87,158 60,680 87,158
Uncertainty ␴ ϭ 0·10
2 0·0000 1,050 51 1,050 51
5 0·0017 1,195 249 1,193 249
10 0·0210 1,485 706 1,454 691
25 0·0721 3,040 3,190 2,821 2,960
50 0·1060 10,250 14,598 9,164 13,051
100 0·1369 112,496 176,300 97,095 152,164
Uncertainty ␴ ϭ 0·20
2 0·0025 1,050 54 1,047 54
5 0·0818 1,212 338 1,113 310
10 0·1969 1,629 1,144 1,309 919
25 0·3425 4,595 6,653 3,021 4,374
50 0·4676 27,134 47,647 14,446 25,367
100 0·6244 879,127 1,618,435 330,200 607,884
Uncertainty ␴ ϭ 0·40
2 0·0709 1,050 114 976 106
5 0·2993 1,303 764 913 535
10 0·4534 2,155 2,765 1,178 1,511
25 0·6260 12,857 26,986 4,809 10,093
50 0·7757 302,365 688,523 67,820 154,436
100 0·9189 155,449,265 382,219,535 12,606,935 30,998,004
Uncertainty ␴ ϭ 0·60
2 0·1612 1,050 232 881 195
5 0·4231 1,418 1,306 818 754
10 0·5699 2,852 5,080 1,227 2,185
25 0·7373 34,627 88,469 9,096 23,241
50 0·873 3,375,768 9,189,769 428,722 1,167,101
100 0·9716 19,177,053,613 59,514,142,750 544,628,323 1,690,201,654
Monopoly Power, which can be expressed as the ratio of value added to the wages of
production workers. Their difference is the cost of the ﬁrm’s non-production activities
and the gross proﬁt on its capital. With a DMP ϭ 1 meaning that there is no monopoly
power, the DMP in the US manufacturing sector ﬂuctuated in a narrow range, around 2·5
from 1899 to 1949, and then rose to 5.25 by 1996 (see Gordon, 1998). The purpose of
these expenditures in the pursuit of monopoly power is to raise the mean and reduce the
variance of the return on capital.
The contrast between this theory of the ﬁrm and mainstream theory is also reﬂected in
the literature that deals with growth. Solow (1970) has explored many interesting
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Capitalism’s growth imperative 43
questions involved in going beyond a two-factor model in which labour and capital are the
only factors of production, the growth rate in wages, proﬁts and output converge, and the
excess of the growth rate in output over the growth rates in labour and capital is due
to exogenous technological progress. The current new idea in growth theory is the distinction
between exogenous and endogenous growth, the former (latter) being generated
by institutions and policies outside (inside) the economic system. However, how this
endogenous growth takes place is far from clear, apart from being the inevitable fruit of
government support for education and research. It does not take place in ﬁrms where
agents implement or subvert technologically given production functions (see Romer,
1994; Solow, 1994; Nelson, 1997). The contrast between the two theories of the ﬁrm
is further illustrated by the literature on globalisation, where the anti-establishment literature
sees corporations ruling the world with terrible consequences for people and the
environment (see Barnet and Cavanaugh, 1994; Greider, 1997).
We have seen in Table 4 how a ﬁrm can achieve both survival and growth well into the
future. It must engage in a high rate of net investment, make the gross proﬁt on production
greater than the sum of the expenditures on administration, other non-production
activities, investment and dividends. And it must also enjoy a low variance in the rate of
return on capital. Successful expenditures in the pursuit of monopoly power make all this
possible for any one ﬁrm, and for a sub-group of ﬁrms in a system. But how is it achieved
for an entire system? Expanding the role of government is one means of achieving the
goal, but the objective of capitalist growth is not to ﬁnance a welfare state. In fact, the
elimination of the welfare state is now widely seen as the ideal growth policy.
Perhaps we should close with a puzzle that starts with the same statistic as the ‘equity
premium’ puzzle discussed in footnote 1 on p. 34. All corporations traded on the NYSE
over the long period 1926–90 had a real arithmetic mean rate of return on their shares of
8·6%. The rate of growth in GNP over these years was more like 3%. Technological
progress and improved training of the labour force help explain the growth in GNP and
the stock market, but not the higher growth in the stock market. Is the Luxemburg thesis
mentioned earlier part of the explanation? What else is at work?
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Appendix
In this Appendix, we prove various results claimed in the text.
We begin with a general result which says that our model only depends on seven particular combinations
of parameter values.
Theorem 1
Consider the model presented in the text as equations (1)–(4), (5a), (6a) and (9). The distributions
of K(t) and M(t), for all t Ն 1, depend on the various parameter values only through the seven
quantities ␴, r, h, y, ␥, (g – 1)␭, and ␮ – ␩ – g␭. That is, if the individual parameter values are
changed in such a way that these seven quantities are unchanged, then the distribution of K(t) and
M(t) is unchanged for all t Ͼ 1.
Proof
We can write ␣(t) ϭ ␮ ϩ ␴N(t), where N(t) are standard normal random variables (i.e., with mean 0
and variance 1).
In terms of this, we compute algebraically that
K(t ϩ 1) ϭ K(t)[1 ϩ (g–1)␭] ϩ h max[SP(t),0] (A1)
and
M(t ϩ 1) ϭ M(t)[1 ϩ r] ϩ K(t)[␴N(t) ϩ ␮–␩–g␭]–(y ϩ h) max[SP(t),0] (A2)
We thus see explicitly that these equations can be written entirely in terms of the seven stated
quantities, which completes the proof. a
We now turn to issues of bankruptcy. We require some preliminaries.
We shall have occasion to consider continuous-time versions of the processes K(t) and M(t),
denoted KK(s) and MM(s), respectively. To deﬁne them, ﬁx all the model parameter values and an
integer t Ն 1. Then, given values of K(t) and M(t), ﬁnd the (deterministic) value K(t ϩ 1) and the
distribution of M(t ϩ 1) of the form Normal(m,v), both speciﬁed by our model. In terms of all of
this, we deﬁne KK(s) and MM(s) for t Յ s Յ t ϩ 1, in differential form, by
KK(t) ϭ K(t)
MM(t) ϭ M(t)
dKK(s) ϭ [K(t ϩ 1)–K(t)]ds
dMM(s) ϭ [m–M(t)]ds ϩ Sqrt[v]dBs
here {Bs} is standard one-dimensional Brownian motion.
These continuous-time processes KK(s) and MM(s) have been deﬁned precisely so that KK(t) ϭ K(t)
and MM(t) ϭ M(t) for every integer t. Indeed, these processes simply extend our discrete-time
processes to continuous time. (In the language of stochastic processes, we have embedded our
discrete-time processes into continuous-time processes.)
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46 M. J. Gordon and J. S. Rosenthal
Using this continuous-time embedding, we give a precise deﬁnition of ‘bankruptcy’ as we shall
prove it.
Deﬁnition
We shall say that ‘bankruptcy is certain’ for a model, if the following holds: With probability 1, either
KK(s) ϩ MM(s) is negative for some s Ͼ 1, or else KK(s) [and hence also K(t)] converges to 0 as s
(and t) go to inﬁnity.
In terms of this deﬁnition, we have the following.
Theorem 2
Consider the model presented in the text as equations (1)–(4), (5a), (6a) and (9).
Suppose that the model parameters are such that for all t Ն 1, the conditional expected values of K(t)
and M(t) satisfy that
E{[K(t ϩ 1) ϩ M(t ϩ 1)]–[K(t) ϩ M(t)]|K(t),M(t)} Յ 0 (A3)
Suppose further that r ϭ 0 and ␴Ͼ0. Then bankruptcy is certain [regardless of the initial values K(1)
and M(1)].
Proof
Let KK(s) and MM(s) be the continuous-time versions of K(t) and M(t), as above. Let
␶ ϭ inf{s Ͼ 1; KK(s) ϩ MM(s) Յ –1}
Deﬁne a new process {Xs} by
Xs ϭ ΆKK(s) ϩ MM(S), ␶ Ͻ s
Ϫ1, ␶ Յ s
(In particular, note that Xt ϭ K(t) ϩ M(t), for any integer tՆ1.) That is, Xs is the total wealth of the
player at time s, except that {Xs} gets ‘stopped’ at –1 as soon as the player’s total wealth hits this
value.
In terms of {Xt}, equation (A3) says that E[Xt ϩ 1–Xt|K(t),M(t)]Յ0. Because of the construction of
{KK(s)} and {MM(s)}, it follows from this that E[Xs ϩ r–Xs|Xu (u Յ s)] Յ 0 for any r Ͼ 0. That is, the
process {Xs} is a supermartingale, i.e., on average it will stay constant or get smaller.
In particular, E[Xs] Յ X1 ϭ K(1) ϩ M(1) for all s. On the other hand, since {Xs} was ‘stopped’ as
soon as it hits –1, we see that Xs Ն –1 for all s Ͼ 1.
We conclude that {1 ϩ Xs} is a non-negative supermartingale with bounded expectation. It then
follows from the standard Martingale Convergence Theorem (see e.g., Theorem 14.2.1 of Rosenthal,
2000) that with probability 1, the sequence {Xt} must converge pointwise to some limiting
random variable, say X.
Now, if X ϭ –1, then in particular Xs Ͻ 0 for some s. This implies that MM(s) ϩ KK(s) Ͻ 0. Hence,
in this case the player was bankrupt before time s.
On the other hand, if X Ͼ –1, then {MM(s) ϩ KK(s)} converges to X, hence also {M(t) ϩ K(t)}
converges to X. In this case, (Kt ϩ 1 ϩ Mt ϩ 1) – (Kt ϩ Mt) converges to 0. But from equation (A2)
above, we see that conditional on Kt and Mt, the difference (Kt ϩ 1 ϩ Mt ϩ 1) – (Kt ϩ Mt) has
conditional variance [␴K(t)]2
. If the difference converges to 0, then this variance [␴K(t)]2
must
converge to 0. Since we are assuming that ␴ Ͼ 0, then this implies that K(t) converges to 0. Hence, if
X Ͼ –1 then K(t) converges to 0.
Since we always have either X Ͼ –1 or X ϭ –1, this completes the proof of the theorem. a
Theorem 3
Consider the model presented in the text as equations (1)–(4), (5a), (6a) and (9).
Suppose that r ϭ 0, ␴ Ͼ 0, and that ␮ Յ ␭ ϩ ␩. Then bankruptcy is certain [regardless of K(1) and
M(1), and regardless of the values of g, h, ␩, y, and ␥].
Proof
Again let Xt ϭ K(t) ϩ M(t) be the total wealth of a given player. Then
CJE 27/1 pp. 025-048 REV 13/11/02 3:00 pm Page 46
Capitalism’s growth imperative 47
Xt ϩ 1–Xt ϭ –␭K(t) ϩ ␣(t)K(t)–␩K(t)–y max[0,SP(t)] ϭ [␣(t) – ␭ – ␩]K(t)–y max[0,SP(t)].
Recall now that ␣(t) has mean ␮, so that E[␣(t) – ␭ – ␩] ϭ ␮ – ␭ – ␩ Յ 0. Furthermore max[0,SP(t)] Ն 0.
Therefore,
E[Xt ϩ 1–Xt|K(t),M(t)] Յ 0
i.e., {Xt} is a supermartingale.
The result now follows from Theorem 2. a
Remark: Note that this Theorem remains valid for any value ␤ Յ 1, not just ␤ ϭ 0·5.
Remark: Suppose ␮ ϭ g␭ ϩ ␩. In this case, if g Յ 1, then the conditions of Theorem 3 are satisﬁed
and eventual bankruptcy is certain. However, if g Ͼ 1, then the conditions of the theorem are not
satisﬁed, and it may be possible to never go bankrupt, or at least to postpone bankruptcy longer.
Remark: This theorem can be interpreted informally as saying that, if the parameters satisfy the given
conditions, then bankruptcy is certain in the long run. Strictly speaking, our conclusion of
bankruptcy must also allow for the possibility that a player will keep all their wealth in cash M(t) Ͼ 0
even as their capital wealth K(t) goes to 0. However, this possibility can be ruled out of g Ն 1, as the
following theorem shows.
Theorem 4
If the hypotheses of Theorem 3 are satisﬁed, and if furthermore g Ն 1, then there is s Ͼ 1 with MM(s)
Ͻ – ␤KK(s) for any ␤ Յ 1.
Proof
If g Ն 1, then K(t) Ն K(1) for all t Ն 1. It is therefore not possible that K(t) converges to 0 in this case.
According to Theorem 3, the only other possibility is that there is s Ͼ 1 with MM(s) ϩ KK(s) Ͻ 0.
The result follows. a
We now consider further the ‘no growth condition’ discussed in the text, namely
(g–1)␭ ϩ h␮ ϭ ␮ – ␩ – g␭ – ␮(y ϩ h) ϭ 0 (10)
If y ϭ h ϭ 0 then this equation implies that ␮ ϭ g␭ ϩ ␩; hence, if g Յ 1, then the conditions of
Theorem 3 are satisﬁed. However, if y and/or h are positive, the situation is less clear. We have the
following.
Theorem 5
Consider the model presented in the text as equations (1)–(4), (5a), (6a) and (9).
Suppose we replace max[SP(t),0] with SP(t) in equations (5a), (6a) leaving equations (1)–(4) and
(9) unchanged.
Suppose further that (10) holds and that r ϭ 0. Then for this modiﬁed model, E[K(t)] ϭ K(1) and
E[M(t)] ϭ M(1) for all t Ͼ 1 (i.e., on average there is zero growth). Hence, if ␴ Ͼ 0, then Theorem 2
applies, and bankruptcy is certain.
Proof
We prove by induction that E[K(t)] ϭ K(1) and E[SP(t)] ϭ ␮K(1). Recall that SP(1) ϭ ␮K(1),
hence the statement is obviously true for t ϭ 1.
Now assume it is true for some t Ն 1. Then from equation (A1) above, with max[SP(t),0] replaced
by SP(t), since (g – 1)␭ ϩ h␮ ϭ 0 and E[SP(t)] ϭ ␮K(1), we see that E[K(t ϩ 1)] ϭ K(1). Now, since
P(t) ϭ ␣(t)K(t), and since ␣(t) and K(t) are independent, it follows that E[P(t)] ϭ ␮E[K(t)] ϭ ␮K(1).
But then since
SP(t ϩ 1) ϭ ␴␥SP(t) ϩ (1 – ␥)P(t)
it follows that E[SP(t ϩ 1)] ϭ ␮K(1). Hence, it follows by induction that E[K(t)] ϭ K(1) and
E[SP(t)] ϭ ␮K(1) for all t Ͼ 1.
But once we know that E[SP(t)] ϭ ␮K(1) and (10) holds, then it follows from equation (A2) that
E[M(t ϩ 1)] ϭ E[M(t)] for all t Ն 1. Hence, by induction again, E[M(t)] ϭ M(1) for all t Ͼ 1. This
completes the proof. a
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48 M. J. Gordon and J. S. Rosenthal
We conclude from Theorem 5 that, if (10) holds, then any possible escape from eventual bankruptcy
must occur entirely due to the difference between max[SP(t),0] and SP(t) in our model.
Remark: If we do not replace max[SP(t),0] with SP(t) in equations (5a), (6a) then Theorem 5 does
not apply. However, it is not clear in this case that the probability of bankruptcy is reduced. Indeed,
the simulations presented in the text indicate that eventual bankruptcy is certain in this case as well.
(Mathematically speaking, the ‘max’ in equations (5a) and (6a) slightly increases both I and C. The
increase in C can only lower the player’s wealth. However, the increase in I is more subtle, which
prevents a clear mathematical analysis.)
Remark: Theorem 5 assumes (as do all other results in the Appendix) that the mean proﬁt rate, ␮, is
held constant. If we instead assume mean reversion for ␮, as in equation (12) of the text, then ␣(t)
and K(t) are no longer independent. Hence, Theorem 5 does not apply in this case.
Theorem 6
Consider the model presented in the text as equations (1)–(4), (5a), (6a) and (9).
If (10) holds and r ϭ 0 and ␮ Ն 0 and ␥ ϭ 1, then E[K(t)] ϭ K(1) and E[M(t)] ϭ M(1) for all t Ͼ 1.
Hence again, if ␴ Ͼ 0, then bankruptcy is certain.
Proof
In this case, SP(t) ϭ ␮K(1) Ͼ 0 for all t, so that max[SP(t),0] ϭ SP(t), and Theorem 5 applies
directly. This completes the proof. a
Theorem 7
Consider the model presented in the text as equations (1)–(4), (5a), (6a) and (9).
If (10) holds and r ϭ 0 and ␮ Ͼ 0, then for ﬁxed ␴ Ͼ 0 and ﬁxed ␧Ͼ0, the probability that the player
goes bankrupt or inft K(t) is less than ␧ goes to 1 as ␥ goes to 1. In symbols,
lim P[bankrupt, or inftK(t) Ͻ ␧] ϭ 1, ␥Ǟ1
Proof
As ␥ goes to 1, the variance of SP(t) goes to 0. But SP(t) has positive mean. Hence, max[SP(t),0]
becomes a closer and closer approximation to SP(t). Consequently, the no-growth conditions of
Theorem 5 get closer and closer to being satisﬁed.
Now, write p(␥,␧,t) for the probability that, for a given ␥, the player goes bankrupt by time t, or K(t)
is less than ␧.
From Theorem 5, for any ␦Ͼ0, we can ﬁnd large enough t that p(1,␧,t) Ͼ 1–␦. But then from the
above observation, we can ﬁnd a Ͻ 1 which is close enough to 1 that |p(1,␧,t)–p(␥,␧,t)| Ͻ ␦
whenever ␥ Ͼ a.
It follows from the triangle inequality that p(␥,␧,t) Ͼ 1 – 2␦ whenever ␥ Ͼ a. In particular, for ␥ Ͼ a,
P[bankrupt, or inft K(t) Ͻ ␧] Ն 1–2␦
Since ␦ Ͼ 0 was arbitrary, the result follows. a
Remark: Even if (10) holds with r ϭ 0 and ␮ Ͼ 0, then for large enough ␴ and small enough ␥, there
could be a positive probability of avoiding eventual bankruptcy. However, this probability would
usually be extremely small, since (by Theorem 5) it arises solely because of the very subtle difference
between max[SP(t),0] and SP(t).
CJE 27/1 pp. 025-048 REV 13/11/02 3:00 pm Page 48