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Unformatted text preview: The Momentum Theorem and the Momentum of Money:
A Philosophical Reflection on the Velocity of Money
By Jiang Tao
[email protected] ABSTRACT
There are some common laws in the universe. Economists widely accept the principles of
Newton’s laws of motion. Specific theorems of physics are more eligible analogues to the
expressions of economics. Equation of momentum can be the best analogue of equation of exchanges.
The virgin form quantity theory of money reserves the principles of momentum theorem that is based on
two cornerstones (real quantity and velocity). But modern monetary economists focus mainly on the quantity side. This is a serious violation to the principles of momentum theorem of the
universal objects. To rehabilitate another desolated cornerstone and to reconstruct the
superstructures based on the two cornerstones will be the crux task of the monetary economics
of the 21st century.
In the studies of monetary economics, the author is inspired by some principles of physics.
However, it may be at first suspected by the orthodox scholars if the equations of physics
appear in the economic papers. Therefore the author writes this separate paper to express the
philosophical principles of my studies before the publication of a series papers.
The author believes that in the universe there are some common laws governing both the
movements of nature and the movements of society. The reasons are simple. From the
viewpoint of history, human society is the product of the movements of nature; from the angle
of microcosm, the universe is composed of the same particles. Nevertheless, the evolution of
nature phenomena is much matured than that of social phenomena. So nature scientists hold
great advantages of approaching the common laws of the universe than social scientists do. But
social scientists may take advantages by drawing on the experiences from the nature sciences,
of course not in the simple and mechanical manner but in the philosophical depth.
In this paper, the author tries to introduce the principles of the momentum theorem of physics,
and to analogize the principles of the momentum of money. The analysis begins with the
Newton’s laws of motion, from which the momentum theorem is derived.
I. Principles of Newton’s Laws of Motion1
Though we recognize the status of relative static of objects, actually any object in the universe 1 For the introductory knowledge of physics in this paper, except the quotations from Newton’s writing, I consult with The
New Encyclopedia Britannica 15th edition; and some versions of The Encyclopedia of Physics; and the introductory Physics
Classroom on the Internet (just search for this title in any search engine). is in a status of absolute motion2, no exception in social or economic phenomena. In nature
sciences, the laws that govern the macro movements of the objects of the universe are called
Newton’s laws of motion. 3 Three fundamental principles form the basis of the classical
The first law (Galileo’s law of inertia): An object not subjected to external forces remains at
rest or moves with constant speed in a straight line.4 Modern scientists deem that the first law can be
now regarded as contained in the second law. The second law: The acceleration of an object is directly proportional to the resultant external
force action on the object and is inversely proportional to the mass of the object.5 In the terms
of an equation,
Force = mass × acceleration
In the symbol form, the equation turns to:
F=m × a (1) The third law (the law of action and reaction): If two objects interact, the force that is exerted
by one to another is equal in magnitude and opposite in direction to the force that is exerted by
In spite of their appearance as nature sciences, the Newtonian laws of motion are basically the
common laws of the universe.7 As a matter of fact, economists are keen on applying these
principles to their studies, though unconsciously in most circumstances.
Without question, economic science certainly shares the first law of motion with nature
sciences. Economists firmly believe that behind any change of any economic variable there
must be a certain (or a combination) kind of pushing forces. Otherwise, the variable remains in
its initial status.
How to apply the second law to practice is a complicated task even for the nature scientists.
Firstly, usually there are more than one kind of forces and more than two objects in the
2 “It is a property of rest, that bodies really at rest do rest in respect to one another. And therefore as it is possible, that in the
remote regions of the fixed stars, or perhaps far beyond them, there may be some body absolutely at rest; but impossible to
know, from the position of bodies to one another in our regions, whether any of these do keep the same position to that remote
body, it follows that absolute rest cannot be determined from the position in our regions.” (Isaac Newton, Mathematical
Principles of Natural Philosophy in Robert Maynard Hutchins (editor in chief), Great Books of the Western World,
Encyclopedia Britannica, Inc. 1952, V34, p10)
The Newtonian laws have proved valid for all mechanical problems not involving the objects moving in speeds of light or
the subatomic particles.
“Law I Every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that
state by forces impressed upon it.” (Isaac Newton, ibid. p14)
“Law II The change of motion is proportional to the motive force impressed: and is made in the direction of the right line in
which that force is impressed.” (Isaac Newton, ibid. p14)
“Law III To every action there is always opposed an equal reaction: or, the mutual actions of two bodies upon each other
are always equal, and directed to contrary parts” (Isaac Newton, ibid. p14)
Refer to note 3. interactions. Secondly, the interactive forces may be in different directions and in different
forms. Thirdly, the interactive forces may change their magnitudes, directions and forms all the
time. Fourthly, the interactive objects may absorb or deposit or transform the external forces,
so the accelerations of the impressed objects may not be proportional to the forces in
immediate and direct way (such as the outbreaks of typhoons or earth quakes). But in most
fields nature scientists may reduce the complexity by establishing isolated empirical conditions,
which can only be realized in economics by theoretical assumptions. Therefore, economists
treat the second law of motion in the way more generous than nature scientists. For instance,
for the changes of a dependent variable of the economy, economists make their best efforts to
include any relative independent variable and to consider the complicated transformation
The third law, the law of action and reaction, is also widely accepted by economists who would
pay much of their attention to the interactions between the dependent and independent
variables in their studies. For example, nowadays, hardly can anyone find the once drastic
debate: “does money matter or not?” For we have known that the two variables are interactive
to each other.
The Newton’s laws of motion are the general principles of the mechanics. These principles
have been extended to various concrete theorems that are more eligible analogues to various
economic expressions than the general Newtonian principles. The momentum theorem is the
one among these analogous theorems.
II. The Momentum Theorem of Physics
In the universe, all objects possess mass;8 all objects are in the status of (absolute) motion. A
physics term momentum is defined as “mass in motion”. So any object in the universe must
possess certain amount of momentum in a specified time. The amount of momentum that an
object possessed in a specified time is dependent upon two variables: mass and velocity.9 In
terms of an equation, the momentum of an object is equal to the mass of the object times the
velocity of the object.
Momentum = mass × velocity
In physics, the symbol for the quantity momentum is "p"; thus, the momentum equation can be
p=m × v (2) Where m represents mass and v represents velocity. Equation (2) demonstrates that momentum
is directly proportional to an object's mass and its velocity. 8 “DEFINITION I The quantity of matter is the measure of the same, arising from its density and bulk conjointly. …It is this
quantity that I mean hereafter everywhere under the name of body or mass.” (Isaac Newton, ibid. p1)
“DEFINITION II The quantity of motion is the measure of the same, arising from the velocity and quantity of matter
conjointly (late scientists replaced the words ‘quantity of motion’ with the concise term ‘momentum’, author added).” (Isaac
Newton, ibid. p1) Equation (2) is an outgrowth of Newton’s second law of motion. From this law it follows that,
if a constant force acts on an object for a given time, the product of the force and the time
interval is equal to the change in the momentum. When combined with the definition of
acceleration (a = ∆v / t), the second law results in the following equalities:
F=m × a=m × ∆v
F=m × ∆v
t If both sides are multiplied by the quantity t, the above equation turns to
F × t = m × ∆v (3) In physics, the quantity F × t is known as impulse (I). Since the quantity m × v is the
momentum, therefore the quantity impulse ( m × ∆v ) should be the change in momentum. So
the equation now says that
Impulse = Change in momentum
I = ∆p = m × ∆v (4) Now, let’s compare equation (2), the equation of the momentum, with equation (3) and (4), the
equations of impulse, below:
p=m × v
I = ∆p = F × t = m × ∆v
We see from the above two equations that the ultimate total momentum of an object after a constant external
force acts on it for a given time, should be the sum of p and ∆p : p + ∆p = m × v + m × ∆v = m × ( v + ∆v ) Or
p’ = m × v’ Where, p’ = p+ ∆p ; v’= v + ∆v Now, in the status of a reestablished equilibrium by the end of the given time, the object obeys the first law
of motion, it moves with a constant speed v’ in a straight line.
The impulse-momentum theorem contains four kinds of variables: time, mass, velocity (or ∆v ) and
momentum (or ∆p ). For an object moving in a constant speed, no external force acts on it, so the ∆v
= 0 ), only three variables can be observed. In that case, an equation
of momentum can illustrates the relations of the three variables: p = m × v; or for a status of reestablished
equilibrium: p’ = m × v’. acceleration is zero (F × t = 0; a = III. A Philosophical Reflection on the Velocity of Money
In monetary economics, the equations of exchanges can be the best analogues of equation of
momentum. If we regard the real quantity of money (M/P) as one variable, then the equations
of exchanges would have three variables, exactly the same shape as that of equation of
momentum. Moreover, We should confirm that the variables in equations of exchanges are
really compatible with that of the equation of momentum.
Like any other object of the universe, money is in the status of absolute motion. Taking the
financial system as a whole, no matter what positions the individual participant takes, money
never stops moving. To understand money in motion, we must inspect its quantity, quality and
velocity. The quantity of money is its nominal amount M; the quality of money is its
purchasing power adjusted by price index P; and the velocity of money is its circulating rate V.
In monetary economics, M/P is regarded as the real quantity of money ( Mr), quality has been
implied by this expression. So Mr is compatible with the concept of mass that that is the
combination of quantity and quality in physics.10 A thing that be called money must be the
integration of its real quantity and velocity. The product of real quantity and velocity is then
compatible with the concept of momentum in physics. Consequently, the equations of
exchanges of the two traditions11now take the forms of
Y = Mr × Vy (5) T = Mr × Vt (6) Where in equation (5) and (6), Y = nation income; T = total transactions; Mr = real quantity of 10 “The quantity of matter is the measure of the same, arising from its density and bulk conjointly.” (Isaac Newton, ibid. p1)
For the two traditions of the equation of exchange, refer to Bordo (1989). Fisher follows Cantillon (1735) the tradition
viewing money as “in motion”. In contrast, Pigou (1917), Marshall (1923), Keynes(1936), Baumol (1952), Tobin(1956) and
even Friedman (1956) follow Locke(1691) and Hume(1792) the tradition regarding money as “at rest”. So the “rest” tradition
has been extended extensively in the last century. Nonetheless, the “motion” tradition stagnated after Fisher.
11 money, Vy = income velocity of money, and Vt = transaction velocity of money.
We encounter a crucial problem here: which of the two equations is more compatible with the
equation of momentum? To solve this problem, let us first focus on the variables of the
equation of momentum (p = m × v).
All of the three variables in equation of momentum are authentic and observable; and the
momentum p is definitely a dependent on m and v; thus p is rightly the product of m times v.
This relation can be reversed according to Newton’s third law (action and reaction). The
reciprocal can be explained by the equation of impulse (I=F × t=m × ∆v ), in where, under the
condition of normal speed, m would not change, ∆v is determined by the changes of
momentum (I = F × t).
Now we turn to equation (5). At first sight, Vy, the income velocity of money, is quite attractive.
Nevertheless, in most cases we know that, income transactions are mixed with other
transactions. This is right the way to keep the economy vigorous. On the other hand, it is
impossible to separate the volume of income transactions from the volume of other
transactions. Therefore Vy is not directly observable. In fact, Vy is derived merely through Y
over Mr. Therefore attempting to compute Y through Mr times Vy is no difference from arguing
in a circle. Though some scholars may argue that the velocity of money should be constant in
the short-run, that argument is at most a hypothesis, how can they provide with us the valid
evidence that based on the observable data? For no possible in anywhere or in anytime may we
find the direct and reliable data for Vy. To sum up, Vy is not an independent variable; and no
direct data available to establish an authentic Vy; and equation (5) is arguing in circle logically.
Due to these fundamental defects, it is hardly to say that Y is the dependent variable of Mr and
Vy. The reciprocal (required by the third law) is also untenable therefore. In spite of the
similarities in appearance, the income version equation of exchange is not compatible with the
equation of momentum.
It is the turn of equation (6). At first glance, it seems that equation (6) is inferior to equation (5).
Firstly, money is composed of different components; some components are hard to be traced in
economic transactions (such as banknotes). Thus it is difficult to calculate the weighted
average velocity of money. Secondly, relative to the former shortcoming, the total transaction
volume T is hard to collect. Thirdly, even T is available; T includes unnecessary intermediate
transactions and pure speculative financial transactions that are less pertinent to income
But, the merits of equation (6) are just rested on these variables. Firstly, in spite of the hardness
of collecting T and Vt, these variables belong to the firsthand authentic data. Secondly, T is the
dependent variable of Mr and Vt; and no arguing in circle exists there; and the reciprocal is
tenable too; a changing T requires the changes of Mr or Vt or both. Thirdly, the expectation of
separating income transactions from other transactions is the same as the expectation of
separating pure nutrition flow in our blood vessels. There are plenty rooms for us to study the
correlation between T and Y, or more accurately at present between the approximate T and Y.
Under current technical conditions, we can grasp a respectable proportional or even the approximation of the total volume of transactions. We need not to worry about the omitted
transactions curried by banknotes or other less liquid financial assets. We may pay our major
efforts to deal with the volume of transactions of the transaction accounts, for there accounts
are the dominating instrument of the social payment system. In addition, the Fed has a long
history and a lot of experience in collecting the debits of major transaction accounts.12
More importantly, from the evolution of institutional and technological point of view, it would
be absolutely certain that we can observe and supervise the whole economic transactions in the
near future; so that T and Vt (and their subdivisions) would be precisely calculated; the
momentum of money would be automatically presented to the experts as well as to the public.
The correlation between T and Y would be entirely understandable and predictable. That
statistical system would be equally valuable to the statistical system of income-expenditure in
the future. The unbelievable task today would be easily conducted by the developed payment
system and by the advanced colossal computer system tomorrow. So the author believes that
equation (6) is a promising model that its variables are fully compatible with the variables in
equation of momentum.
In my viewpoint of philosophy, the universe is noting but the interactions of the moving
objects. Thus, the principles of momentum theorem are applicable to economic movements
too. 13 Quantity theory of money in its virgin form reserves the principles of momentum
theorem that is the extension of the Newtonian laws of motion.
Specifically, the quantity theory of money is based on two cornerstones. One is the real
quantity of money; another is the velocity of money. The two cornerstones are exactly the
analogues of the two independent variables of momentum (mass and velocity). Once the two
cornerstones are being separated, the “real” quantity theory is doomed to collapse.
Unfortunately, the 20th century has seen such a scenario. As we have known that equation (5) is
not a sustainable model. It was merely a transition form of the Cambridge’s cash balance
equation that is inherited by Keynes. As for equation (6), Friedman and the so-called
monetarism also left it in the cold. Nowadays, no one is serious to the congenital deficient
income velocity of money; no one may recall the transaction velocity of money. Therefore, the
focus of modern monetary economists is mainly on the quantity side, either on the real quantity
of money or on the transformed and mixed quantity of money (the flows of the income identity
are such mixtures).
The collapse of the real quantity theory of money does not mean its fall. Negation of negation
is another common law of the universe. As the author implies above, the quantity side has been
inherited and transformed by different schools; yet, the velocity side is desolated continually.
This is a serious violation to the principles of momentum theorem of the universal objects. The
revival of the real quantity theory relies on both of the two cornerstones. To rehabilitate 12 Unfortunately, the Fed had stopped to collect and publish the debits data since September 1996. Refer to my paper, “The
Mismatch of Fisher and His Equation of Exchange: A Proposal to the Federal Reserve System”
Momentum theorem is derived from the second law of motion; therefore the restrictions to Newtonian laws are valid to this
theorem. Refer to note 3 again. another desolated cornerstone and to reconstruct the superstructures based on the two
cornerstones will be the crux task of the monetary economics of the 21st century. ...
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