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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 Nankai University [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 Newtonian mechanics. 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 the later.6 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) 3 The Newtonian laws have proved valid for all mechanical problems not involving the objects moving in speeds of light or the subatomic particles. 4 “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) 5 “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) 6 “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) 7 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 mechanisms simultaneously. 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 rewritten as 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) 9 “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 t Or 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 Or 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 t 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 transactions. 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” http://papers.ssrn.com/author=254805 13 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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This note was uploaded on 10/24/2011 for the course SCIENCE PHY 453 taught by Professor Barnard during the Winter '11 term at BYU.

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