Dr.
THE ECONOMY AS A DYNAMICAL
SYSTEM
-
written in 1995 based on a talk given by the author that year as part of the
Ventures in Research series at C.W.Post Center of Long Island University
All
of us are familiar with the liberating thesis that Thomas
Jefferson wrote into the
Declaration of Independence of our country.
The only just purpose of government is to protect the
rights of the people, and therefore
"Whenever any form of government
becomes destructive of these rights,
it is the right of the people
to alter or abolish
it." [1] Moreover,
notes Jefferson, although it is also true that people should
not abolish governments "for light and transient causes", the people are not in fact prone to do so. Indeed,
it is their right and their duty
to throw off governments that have evinced a design to reduce them under
absolute despotism.
Less familiar to most of us is that Thomas Jefferson,
noting that "persons and
property make the sum of
the objects of government" [2] repeatedly throughout his life applied a
similar thesis to the question of property.
In these writings, he views
property as distributed to particular people for the sake of the encouragement of
industry. But when this
distribution becomes destructive of
this purpose it is the right of
the people to redistribute it. (See e.g. [3],[4],[5] and
[6])
Below
is an excerpt from a letter written by Thomas Jefferson regarding the
economic conditions to which he was witness in 1785 during his stay in
" The
property of this country is absolutely concentrated in very few hands, having revenues of from
half a million of guineas a year downward.
These employ the flower of the
country as servants, some
of them having as many as 200
domestics, not labouring. They employ also a great number of
manufacturers, and tradesmen, and
lastly the class of
labouring husbandmen. But after all there comes the most numerous
of all the classes, that is, the poor who cannot find work. I asked myself what could be the reason
that so many should be
permitted to beg
who are willing to work, in a country where there is a very
considerable proportion of uncultivated lands? These are undisturbed only for
the sake of game. It should seem then
that it must be because of the
enormous wealth of the
proprietors which places them above
attention to the encrease of their
revenues by permitting these lands to be laboured. I am conscious that an equal division of property is impracticable. But the consequences of this enormous
inequality producing so
much misery to the bulk of
mankind, legislators cannot invent
too many devices
for subdividing property, only taking care to let their subdivisions go
hand in hand with the natural affections of the human
mind. The descent of property of every kind therefore to all the
children, or to all the brothers and sisters, or other relations in equal degree is a
politic measure, and practicable
one. Another means of silently
lessening the inequality of property is to exempt
all from taxation below
a certain point, and to tax
the higher portions of
property in geometrical progression as
they rise. Whenever there
is in any
country, uncultivated lands
and unemployed poor, it is clear
that the laws of property have been so
far extended as to violate natural right. The earth is given as common
stock for man to labour and live
on. If for
the encouragement of industry we allow it to be appropriated, we must
take care that other employment be provided to those
excluded from the appropriation. If we do not the fundamental right to labour the earth returns to the
unemployed." [3]
Today, even more than in the
An updating of both the general philosophy
of Thomas Jefferson and his explanation of the above irony are very much in
order.
This updating is very much facilitated by a
number of factors. One of these
is the very
significant work of the
noted mathematician, Jacob
Schwartz [11]. Another is the fact that the subject of
dynamical systems has become an important field
of mathematics. Finally, the accessibility of the computer screen in
depicting such systems is very important.
Fundamental in
Let E designate this amount of wealth
measured say in francs. Thus, when the
amount of wealth available to the wealthy handful as a
whole becomes greater than E,
they will be placed "above
attention to the encrease of their revenues
by permitting these lands to be
laboured".
Let Q
be on an average the amount of wealth produced by each of the employees of the wealthy handful beyond
the amount that they are paid by their
employers. Thus, Q is the amount of profit extracted by the wealthy proprietors on an average from
each of their employees.
(We will refer to
Q as the
rate of exploitation.)
Let N be the number of these employees.
Then,
the amount of profit made by the
wealthy handful as a whole is apparently
N times Q.
Thus,
if N times Q is less than E the
wealthy handful will employ more
laborers because all the needs of the handful will not
be met, and if N times Q becomes
equal to E, the
wealthy handful will be
sated and will not bother to employ
any more laborers.
The economy, then, is at equilibrium when N
times Q = E.
Dividing
both sides of our equation by
Q, we obtain an equation for the exact amount of workers
that will be employed at the moment when the wealthy handful are sated:
(1) N = E/Q.
Looking
at our equation, we see that the
employment level N falls when
the rate of
exploitation Q is high,
and that therefore, the
way to increase
the employment level in Jefferson's model, barring a change in the habits of the wealthy
handful is for
the workers to demand a greater
share of the wealth they produce.
The
How
does our modern situation compare with the one described by Thomas Jefferson?
In
our time the
means of production,
the mines, mills, factories, farms
etc. are owned
for the most part
by corporations, which are
legally the property of their share-holders
in proportion to their
ownership of stocks.
(Senator Kefauver who had been chairman of the Senate
Subcommittee on Antitrust and Monopoly
from 1957 to 1963
reported that 50 corporations were
making 48% of the profits in manufacturing [12]).
A
very tiny handful
of people, less
than 1.7% of our population, owns
more than 82.4%
of the stock in
these corporations [13], [14]. Thus, in our own time, as in
Moreover,
the rest of the population, for the
most part, cannot make ends meet
except by working for either the
nation's businesses or the government for wages and salaries.
These salaries and wages, in our day as in
One
of the problems with Thomas Jefferson's theory would appear to be that given the
habits of people of our time, there does
not seem to be any particular amount of wealth that will put an end
to the search by the wealthy of
our own time for more wealth.
But this is only true if we think of wealth
in its money form, and in the form of objects that are not consumed by an
individual such as factory buildings etc.
Let us think, instead,
of wealth as the total value of the consumer goods and services in the possession of the
individual measuring that value by adding up the prices of these
(using the prices as they
were in some particular year in
order not to consider the effects of inflation.)
Although
the individual will always want to make more money, he
will not have an infinite
appetite for shoes,
automobiles etc., i.e. for consumer's goods.
As
the noted economist John Maynard Keynes[20] pointed out, the wealthy do not increase their purchases of
consumer's goods in proportion to the rise of their incomes. At a certain point, their appetites are, indeed, sated, and they seek at that point
to save
their wealth or invest it in order
to increase it. (Indeed, the possession by wealthy
individuals of money that they do not feel
any need to spend will have a tendency to cause the prices of the things they
buy to rise.) Moreover, only a portion
of the profits of the corporations are in fact distributed to the stock holders
for their consumption.(e.g. see [21])
Thus,
by E we will not mean the amount of wealth in the form of money, bank accounts, stocks in corporations etc.
but rather E will be understood to be the total value (as
measured in the prices of
a standard year) of the
consumers' goods that
the wealthy handful purchase in the course of a "day". (Here we will use the term "day"
simply to mean a convenient unit of time.)
The second problem with our equation is that
it considers only consumers' goods. Obviously,
there are other goods
such as machinery, factory buildings, lubricating oil, the leather
to make shoes etc. that constitute what economists call
producers' goods. These goods will in
general be the property of business
firms that are possessed in one form or another for the most part by the
wealthy handful. In addition these firms will be
in possession of unsold consumer and producers' goods, some of the latter
of which will be in use in production. We will refer to the total value, then, in the possession
of these institutions as their
inventory, and we
will indicate the amount
of this inventory at the
beginning of each "day" t respectively by I(t). The net increase in
inventory of the nation's business firms in a particular day t will then be the
difference
I(t+1)-I(t).
Samuelson
refers to this quantity as "net investment" [22],
What we call the net profit, then, to the nation's businesses will be,
E + I(t+1) - I(t).
i.e. The
net profit is the amount spent by the handful out of
profits for their
consumption plus the amount
of growth in inventories (the latter of which, of course, will be negative in times in
which old machinery is being
permitted to run
down without being replaced, and in which inventory is being sold off
without replacement).
Leaving
out of consideration the effects
of foreign trade, there
is only one
other important part of the net national product of the nation for the
year, namely the the value of the goods
and services purchased by the workers who work for
the nation's firms.
For
the sake of simplicity we will
assume that there is a daily after
tax wage w paid to each employed worker,
and that this wage is wholly
spent on consumers' goods. Obviously, this is somewhat distorted
by the existence of
buying on credit, mortgages etc.,
small amounts of savings, and the
existence of different wages paid to
workers in different
occupations.
Nevertheless, I
think that it
is revealing to make
this oversimplification.
(According to
We
will also assume
for the sake of simplicity that
the quantities E, G and
Q are constant. This, of course
is an oversimplification. Q
should be expected to be ever
increasing with technology.
Moreover, it should be expected
to fall during recessionary periods when there is a less efficient use of
labor, and a delay in the replacement of old plant, machinery
etc. E, also, should be expected
to respond to some extent to increase or decrease of
profits, but it is
restricted by the Keynesian-Jeffersonian phenomena we have already discussed. G is also not quite constant.
On the other hand these
quantities are
approximately constant as compared
e.g. to the
growth of inventories
I(t+1)-I(t)
during a business cycle.
Moreover, I think it is revealing
as a first approximation to
reality, and also as a means of suggesting
policy changes that can improve the human
condition, to see what happens if
indeed these quantities are constant.
Thus,
if N(t) is the total number of
workers employed on the "day" in question by the nation's firms, N(t) times w is the after tax wages paid, and
the net national product is:
(3) N(t)*w + I(t+1) - I(t) + E + G. (We are
using the symbol * to indicate multiplication.)
If Q
as in the
(4) N(t)*Q = I(t+1) - I(t) + E + G.
Thus,
we have returned to the original
Jeffersonian equation with the addition
of G and the part of the product I(t+1)-I(t) that remains in the hands of the
business firms.
Note that N(t)*(Q+w) is the entire product
of the nation, and that Q+w is then the product per worker, a quantity that
tends to grow as new technology is introduced.
(In the years from 1948 to 1973
the output per hour for workers in nonfarm business grew at an annual rate of 2.5%. After that period it has grown at the
slower rate of 0.7% in the
Dividing
both sides of (4) by Q, we obtain
the equation for the employment level:
(5) N(t) = ( I(t+1) - I(t) + E + G )/Q.
Thus,
the employment level depends not only on the relatively stable
quantities: E, Q, G but the varying quantity
I(t+1) - I(t) (i.e. the net
investment).
We
will call the value of N(t) when
net investment is zero, equilibrium employment and designate it by Nequi.
(6) Thus, Nequi = (E + G)/Q.
When net investment I(t+1)-I(t) is positive
the inventories of the nation are growing and employment is higher than Nequi.
When I(t+1)-I(t) is negative (i.e.
inventories are wearing out or
being sold at
a faster rate
than they are replaced),
employment is less than Nequi.
When
employment is exactly Nequi
the economy reproduces itself
exactly each year. Conversely, when N(t) < (E+G)/Q, more is being consumed than produced and the
economy contracts, and therefore inventories fall; when N(t)
> (E+G)/Q inventories will
grow. From
this fact alone it should be clear
that employment cannot remain
permanently above equilibrium without
inventories growing out of all bounds.
Solving
equation (5) for I(t+1) we get our
first dynamic equation:
(5')
I(t+1) = I(t) + N(t)*Q - E - G.
We will need one more equation, giving
N(t+1) in terms of I(t) and N(t) in
order to have a dynamical system expressing N and I on each "day" given their state a
"day" before.
In to order complete our model, then, we need to consider the
motivations for the choice of N(t+1) by the nation's business firms i.e. the number of workers
they are to employ.
Dr.
Schwartz [11], in his model,
makes the assumption, based on an analysis of marketing considerations generally,
and in particular a study of the leather-hide-shoe
industry, that each of the nation's businesses
wishes to have on hand, respectively,
an amount of inventory equal to a particular number
of day's sales of its
product. Obviously, too little on hand will leave the business
unable to meet orders that come in, and
too much on hand will create the
possibility that a whole number of
factors including changes of styles, changes of methods of production
(if the articles in question are means of production), and indeed the fact that
it is inevitable that every upturn of the
economy is
followed by a downturn (there have been 8 cycles since
World War II [16]) will
convert unsold inventory into trash.
Indeed, improvements in the methods
of production of the
inventory itself, are apt to cause a fall in its price.
We
will designate by c the average number of days sales that the businesses wish to have in stock.
Sales
to the government (or recipients
of government wages etc.) are given by G. Sales to the
wealthy handful are given by E. Sales to
the workers are given by w*N(t). Thus,
E
+ N(t)*w +
G is the total
amount of consumer and government sales.
There are also sales of producer's
goods, of machinery plant etc. This is a difficult function to estimate,
being responsive to a great many factors.
But at the very least, we cannot expect the amount of these goods that
are purchased to grow to more than some
multiple of the maximum number of
workers employed in the country(who, after all, are expected to
interact with these goods).
For simplicity, I will assume that the purchases of equipment
are given by the simplest possible increasing function of N(t)
(7)
a*N(t) + b where a and b are constants.
This
function of N can be replaced by any increasing function of N without
affecting our results qualitatively. On the other hand, it should
be noted that the advance of technology
should be expected to give us a different such function in
every economic cycle, and that contrary to the very regular graph
of the cycles of the economy shown in figure 1,
we should expect each cycle to be somewhat different from the last.
Thus, the total number of sales is given
roughly by:
(8) s(t) = E + G + w*N(t) + a*N(t) + b.
Thus,
the target inventory of our nation's firms for the t+1 st year will be:
(9)
c*(s(t)) = c*(
If
I(t) is less
than this quantity our firms will seek
to increase production in order to increase their inventory. If I(t)
is greater than this quantity they will seek to cut
down their inventory.
Thus,
there will be a desire to employ on "day" t+1 a number N(t+1)
of workers which the firms calculate will raise (or lower) inventory to that
level by the "morning" of "day" t+2.
Their
expectation of the inventories
for that time will be that by
employing t+1 workers they will
add to the inventory I(t+1) an amount N(t+1)*(w+Q)
i.e. the number of workers N(t+1) to
be employed multiplied
by their productivity
w+Q. The businesses will
then subtract from
this their estimate
of "tomorrow's"
sales. Estimating these by the
amount of the sales of the day
before. Thus, their expectation will be that
the
inventory tomorrow morning produced by employing N(t+1)
workers will be I(t+1)+N(t+1)*(w+Q)-s(t).
Thus, they wish to employ a number N(t+1)
of workers given by:
(10) N(t+1)*(w+Q)-s(t)+I(t+1) = c (s(t))
i.e. solving for N(t+1):
(11) N(t+1) = ((c+1)*s(t) -
I(t+1))/(w+Q).
Replacing s(t) by its expression in
(8) above, and replacing I(t+1) by its expression in (5') in terms of I(t)
and N(t), we
have:
(11') N(t+1) =
(
(c+2)*(E+G)+((c+1)*(w+a)-Q)*N(t)+(c+1)*b - I(t))/(w+Q).
There are technological considerations that
may interfere here. e.g. There may be a shortage of available means of production. (Obviously, a
shortage of money and credit may interfere also. But these will not be considered in
this paper.)
Thus,
in our model we will require that
N(t+1) as given by formula (11')
will be reduced if the inventory
I(t+1) on the "morning" of "day"
t+1 is too small to contain all the
equipment a*N(t+1)+b set in motion by the employment of N(t+1)
workers. (This condition is
clearly too rigid, since a certain
amount of production should be expected to take place without using
the best equipment. However,
the condition can be dropped
without changing the basic
pattern of the resulting
orbits of the economy.)
(12) Thus,
N(t+1) in (11') above will be adjusted downward if necessary if a*N(t+1) + b is greater than
I(t+1).
Clearly,
(5'), (11') and
(12) give us a dynamical system
determining N(t+1) and I(t+1) in terms of N(t) and I(t).
Thus,
starting from any values of I and
N, an orbit will be determined on a
Cartesian axis system in which N is given by
the horizontal axis and I by the vertical axis.
Figure
1 is the
graph determined by the above equations produced by a
computer program with given initial values.
As already mentioned, one of the
reasons for the regularity of the
figure is certainly
our failure to consider the
results of technological changes
(or to
deal with the details
of the motivations of businesses
in determining their purchases
of plant, machinery etc.). It is
well known that in reality, each
economic cycle is quite different from the last.
The employment equilibrium equation N =
(E+G)/Q is a vertical line.
By
letting N(t+1)=N(t) in equation (11') and solving for I(t) we obtain the
equation for the straight line in (N,I) space:
(13)
I(t) = u - v*N(t), where
u=(c+2)*E+G) + (c+1)*b, and v= 2*Q - c*w -
a*(c+1).
Equation
(13) then is the condition that there is momentarily no desire to increase or decrease the number
of workers, since it is calculated
that keeping this number unchanged
the target inventories will be
acheived.
The equation (13) will be called the
inventory equilibrium.
However, as
the analysis of figure
1 below shows,
this equilibrium is only momentary (since N(t) will normally not be exactly
Nequi when
I(t)= u
v*N(t), and therefore
the inventories will either
be larger or smaller the next
"day" depending upon whether N(t) is larger or smaller than
Nequi).
Figure 1
The two equilibrium lines divide the
employment inventory plane into four regions:
Region I:
In region I, inventory I(t) is
less than
u - v*N(t), and employment N(t) is less than (E+G)/Q.
In this region there will be an impulse to
increase employment because inventories
are less than optimum. On the other
hand, since N(t) < (E+G)/Q, the inventories will continue to fall,
thus creating an even greater impulse than before for expansion
of industry (since I(t+1) will be less than before despite the fact that
N(t+1) will be more than before).
This
is the region then of accelerated industrial growth and possibly of
shortages.
This
condition will continue until finally N becomes greater or equal to (E+G)/Q with the gap between I and
the equilibrium inventory level greater
than ever (as a result of the prolonged
period of less than equilibrium employment).
This then will lead to further expansion of N to a level greater than
(E+G)/Q. Thus, we enter region II.
Region
II: In region
II, we have
employment greater than
equilibrium, and inventories less than equilibrium i.e.
N(t) > (E+G)/Q, and I(t) < u -
v*N(t)
In
this region there will continue to be a tendency for N to rise
since inventory is less than
desired. On the other hand since N is larger than equilibrium, the
inventories will actually be
rising, and in fact will be
rising at least at some constant
rate
(since if N(t)>Nequi then I(t+1)-I(t)=q*(N(t)-Nequi).
Thus,
we are in the region of rising
employment and rising inventory.
If at
some point N ceases to rise (e.g. if N
finally reaches Nmax i.e. full employment), inventories will continue
to rise at the constant rate of q*(Nmax-Nequi) and will
thus ultimately become greater
than c*(a*Nmax+b+E+G) thus,
leading us out of region II.
More
likely, long before
obtaining N=Nmax we
will have obtained the level
where I(t) is greater than or equal to optimum inventory u - v*N(t). Thus,
there will no longer be an impulse to increase
employment N, but on the other
hand since N is greater than equilibrium inventories will continue to grow.
Thus, we will have both inventory and employment above their equilibriums, and we will be in region III.
Region III: In region III, I(t) > u -
v*N(t) and N(t)>Nequi.
Now,
as a result
of the first inequality we will have
an impulse to decrease N(t). At
the same time, as long as N(t) is greater
than Nequi we will continue to have I(t) rising at least at the rate
(N-Nequi)*Q.
Thus,
we are in a region of rising
inventories and falling employment i.e. the beginning of a
recession.
This
will continue until employment declines to Nequi. But since inventories
have been rising throughout
the preceding period I(t) will be greater than optimum, and therefore
N(t) will become less than Nequi. We are now in region IV.
Region IV: Here, N(t)<Nequi and I(t)
> u - v*N(t).
Thus,
inventories decline as a result of the first equation, and employment
declines as a result of the second.
Assuming that there is to be a
recovery, inventories finally cease to be higher than optimum. At this point they continue to fall however
since N(t) is considerably less than Nequi.
Thus,
we are led back to region I where
both employment and inventories are less
than their respective equilibriums.
Region I, then, is the period of recovery
from a recession.
What
has been illustrated
is an oscillation
around an equilibrium.
Figure
2 is a computer printout of a
much more complicated model involving a
multitude of firms based on Dr. Schwartz
model which includes the effects of shortages.
I
have also produced a computer model in which technological improvements
play a role. The net effect of these is, as should be
expected, that the employment
equilibrium decreases since, presumably the
purpose of the technological improvements is to improve efficiency and therefore to raise
Q.
Clearly,
in our models,
there are two questions that are important to the working people. These
are:
(1) how to raise the employment
equilibrium;
(2) how to moderate the effects of the
recession on employment.
The first objective is served by raising
the quantity (E+G)/Q.
The
second might be served by
improving the conditions of working
people to the point where the recessive phase of the
cycle is experienced by the workers as a pleasant vacation rather than a
devastating destruction of the worker's life.
Prescriptively, our equation would suggest
that we should seek to increase E and/or G and should seek to decrease Q. The
raising of E is
a difficult matter as a
result of the
Keynesian-Jeffersonian phenomenon already discussed. Moreover,
it does nothing to improve the
standard of living of the great masses who should expect to see
an improvement in their condition as a
result of their ever-rising
productivity. Thus, we are led
to prescribe a reduction in the
rate of exploitation Q by raising wages
w and/or reducing the work day,
as well as an increase in government
spending G as did J.M.Keynes (hopefully for housing, health, education, the environment, the economic
infrastructure, aid to the poor countries of
the world to
develop their infrastructures
etc. rather than for war). Interestingly enough the quote from Thomas Jefferson at the beginning
of this paper suggests one
method by which the government could
collect the money for
massive federal spending
without issuing bonds
or printing money, while at the
same time encouraging investment: a graduated tax on the total wealth of
individuals and corporations "in geometrical progression" as that
wealth rises exempting "all from taxation below a certain point".
It
is only proper
to add here
that life is far
more complicated than are
mathematical models. The
Keynesian prescriptions were very much in vogue until the early 1970's, some
authors posing such solutions as the
definitive easily achievable answer to the revolutionary socialist
movement (e.g. see the
1973 edition of Samuelson's elementary
economics text book [27]).
But
these prescriptions are
definitely out of style today.
Such
writers as Marx [28] and Mill
[29] had long ago strongly suggested that
there was some limit to the amount of
affluence that could be possible without destroying the wage system,
that system being based upon the fact that the great mass
of people have no
significant share in the
ownership of the
means of production, and
are sufficiently poor so that they can only survive by selling their services to
those who own these means of production.
Mill looked forward to a time
when a more educated and affluent working class would reject the role
of wage-worker and form cooperatives instead. Marx quoted authors of his time who
felt that if workers became too affluent, they would become
difficult to
manage. More recently, Heilbroner
[30] made a similar observation,
noting the ability of an affluent
working class to endure the financial difficulties experienced by
workers involved in long strikes.
With
the revolt of the youth against the establishment in the industrial
countries at the end of the 1960's, the anti-Keynesian theme was
taken up by the leaders of the industrialized world [31],
attributing the revolt "to the relative affluence in which most
groups in the Trilateral societies came to share during the economic expansion of the 1960s",
and prescribing massive cuts in federal budgets as a solution. Since the 1960s, we have all seen the
continuous assaults on the federal budget by our government, allegedly for the
purpose of "balancing the
budget", but in fact
having the effect of very much
increasing the government debt. (e.g.
The Government debt tripled in
the Reagan years, despite the fact that
Reagan's major promise to the nation was to cut the
federal
deficit. [32]) We have also seen changes
in the tax laws very much favoring the
wealthy ([33],[34]). Predictably these policies have resulted in a general decrease in the
standard of living at home and abroad.
Contrary
to Samuelson, these
events suggest that
the abolition of poverty
at home and abroad cannot
be achieved without a determined struggle by the great masses of this country
and the world against the establishment in which the struggle for the
Keynesian-Jeffersonian prescriptions can play a crucial role, and which if
successful will necessarily result in a
fundamental restructuring of our society.