| This article applies a dynamics approach in the research of monetary law of movement under the complex system of social economical operation, and characterizes the movement of money in a social institutional framework during GDP's formation. Assuming that humans'pursuit of the return of their money expenditure is a sensible course of nature, it defines the expression of money circulation velocity, then proceed to deduce the basic differential equation of money circulation. By solving this equation, we can get the expression for a GDP dynamics model. After empirically testing the expression, this article arrives at the solution: GDP and the money in circulation (M0) share the only positive correlation when the monetary financial institution remains unchanged. Keywords: GDP, cash holding balances, monetary financial institution, exogenous moneyGDP is the most important factor in judging a country's economical operation and comprehensive strength. All governments are pursuing the rapid and stable growth of real GDP, which reflects the increase in their country's wealth. The measurement norm of GDP growth is the money, i.e. the real GDP's rapid growth is always accompanied by the rapid growth of the money amount. The increase in wealth should be equal to the increase in money multiplied by the velocity of money movement. Therefore,"MV"naturally becomes an important factor in analyzing a country's GDP growth. The research of M's velocity, that is to say, M's dynamics law, is the key in the research of GDP, meantime it becomes the target of a country's monetary and financial policy.I. IntroductionGDP is the sum of monetary value of a country's final products (final products refer to those which can't be used to produce others). It shows the total monetary value of all the goods and services produced by a country's residents in a certain period of time. This total monetary value can be expressed by"MV", which is the key variable of GDP analysis. In the first volume of"a tract on monetary reform", Keynes sees"MV"as the basic variable in discussing"dynamics of price level", and then uses it to analyze the circulation of money in the areas of finance and business. For a long time, economists have literally misused the word"money circulation velocity". They are actually studying the frequency of money usage, which is totally different from the spatial velocity and direction of money. It means that the money circulation velocity is no more than the corresponding transaction volume in unit money. This is completely irrelevant to the fact that some unit money is not used, for example, the cash holding balances. This frequency index only indicates the frequency of money usage in business, so it won't help the economists to solve economic problems. The index of money frequency is not as attractive as the index of money circulation velocity. In economic research, we seek those indexes that are helpful to our study, while some frequency indexes help little in our analysis. The money circulation velocity is not a scalar, but a vector with direction. Its computation is subject to parallelogram principle of vectors, not arithmetic principles.The social economic system is a complex system, and complex systems are all dynamic systems. Dynamics approaches should be applied in the research. Among the scholars who regard economic operation as complex system are F. Quesnay, Irving Fisher, Joseeph A. Schumpeter and Friedrich Hayek. In Quesnay's article published in"Encyclopedia", he regarded economic activities as the circulation of money, goods and personnel among industrial sectors, similar to blood circulation in human bodies. He brought forward the hypothesis that the flow of money and goods is like seepage flow. In Hayek's opinion, people's rationality is a"fatal conceit"in doing with the complex systems of economic operation. Spontaneous free expanding order should be applied in solving the complex system's inner problems.The target of dynamics study of economic operation in complex systems should be the most common object within the system—money. It is very important to research monetary law of movement, especially the circulation velocity, because it vastly helps setting the GDP model. The contemplation of universality is the beginning of philosophy as well as our economic research. Under social institutional framework, various wealth-goods restrict, influence and promote each other. Together with the social institutional framework, they form a complex seeping field. If we want to research a certain wealth-good's movement, we must first identify it. If we don't clarify our research target, but only focus on a complex system, the frame of reference would be lost. In that case we would not be able to distinguish them, let alone study their law of movement. Different approaches should be applied when researching into wealth-goods with different fluidity. If this thinking method is right, the money quantity theorists who focus on M2,such as Milton Friedman,are not able to give out a scientific description of the money movement law. The same difficulties also hamper the researchers who focus on M1. They also cannot give an accurate description of money movement..In natural science, studies of dynamics play a key role. The research target is the movement law of moving substances in mediums, including the moving substance, the medium and their correlations. The researches of substance's movement law include velocity, the direction of moving, substance's own features, physical features of mediums and their correlations. By studying objects'law of movement, we can achieve the control over their movements. Money dynamics is the research of monetary law of movement in human society's (medium's) institutional framework, including the direction and velocity of money movement, the quantity of money and purchasing power, and the great influence and control that social institutional framework will exert on money movement. The wealth-good system in movement under social institutional framework includes land, natural resources, fixed assets, consumption goods, bond, stock, futures, option, money, and foreign exchange. Their fluidities become greater in order. People have been creating new wealth-goods to fill up the space, in order to satisfy the people's diversified need. At the same time, people are also expanding this wealth-good space to further satisfy people's need. More broadly speaking, wealth-goods include culture, scientific tradition, credit degree, diplomacy, national defense and institutions. Money is the symbol for value of all wealth-goods, i.e. general value representation. This kind of value is a concept or a symbol. Money comes from commodity exchange, but it is more than mere commodity. When developing into international money, they begin to have absolute fluidity. In addition, this value symbol is transferred immediately during the product's trade, i.e. the transference is continuous between two sides. Therefore, money has the feature of continuous fluidity in human society. When studying the velocity of money movement, we should take into account the variation of money in the period of taking-in and giving-out among economic units. We can set up the GDP dynamics model by studying money and their movement under social institutional framework in the whole economic operation system which forms a country's GDP, and thus find the influencing factors and approaches to help set up a country's monetary fiscal policy.II. The ModelA. Descriptive modelThe premise of scientific study is the affirmation of basic facts, which must be recognized unanimously with no opposition. For a long time, economists have ignored that the basic facts must be made"unanimous facts"before they constitute the basis of economists'research. The dynamics study of currency motion is to study the motion law of currency in human society, i.e. the law of GDP's formation.a. Social institute frameworkSociety is consisted of individuals, and individuals are inter-linked by social institute. A basic fact can be drawn through this, i.e. the social institute is distributed continuously in human society. These institutes comprises order institute and market trade institute. The turning point between these two institutes is the borderline of all social organizations. There are mainly three kinds of organizations in human society: families, enterprises and governments. Division of labor constitutes the basis of institute. The gaming between improved efficiency from labor division and harnessing cost from order institute decides the size of organizations. The insides of families, enterprises and governments and their correlations are all consisted of institutes, i.e. the whole society is a framework system. The institutes related to currency circulation are distributed asymmetrically in the social institute framework, so the corresponding influence on currency circulation is different. We can choose at our will an institute unit that can be used to study currency motion features, and call it"economic unit". This economic unit should not be too large, because it would not reflect the institute features of currency motion sensitively; it should not be too small either, because it would not qualify to be a prototype of currency motion institute research. Social institute framework is a very complex system, so we can't describe the correlations accurately, which are mainly represented by institute system. However, the institute system still can not precisely clarify the micro institute phenomena caused by the interaction of wealth-goods and social institute framework. We can precisely describe the institute features of currency motion and accurately clarify all the phenomena caused by the interaction of currency and institutes by defining proper characteristic institute units, thus control the motion of currency.b. Features of currency seepageEconomic units balance between adaptability preference and opportunity cost to determine the best cash possession excess. The currency is transferred in a flash during the trade between economic units. In another word, currency is distributed continuously among the trade links both inside the economic units and between them, just as the institutes permeate through human society. In micro view, the currency quantity reflects wealth quantity; in macro view, the momentum index, i.e. currency quantity multiplied by currency velocity (M0*V), reflect a country's wealth quantity. The currency circulation velocity of cash owned by economic units varies greatly. It also varies greatly in the monetary purchase of commodities and services, real element of production, stocks, bonds and deposit certificate. Permeating throughout human society, currency has different value in different trade links. For example, the currency in trade links and the currency as cash possession excess show different values. In addition, the currency value changes continuously without any intermission in the time scale.The purchasing power of currency varies greatly in different areas and time periods, as well as to different commodities. Under different situations, people's estimates of its value vary a lot. Regarding currency fluid as continuous mediums actually means that the purchasing power, i.e. the value of currency, changes continuously.Similarly, the fluid can also be defined as continuous function of time t. In another word, the purchasing power of currency changes continuously in time scale.c. Correlations between currency and social institute frameworkThe impetus of currency's circulation in social institute framework is no doubt the return of currency expenditure. In order to win public support, the government is inclined to maximize the supply of public goods. Then the currency circulation system in which government put money into the whole economic social system and gets taxation from economic units is formed. In modern society, every country's government applies either expansionary currency policy or deficit financing policy. In these cases, the government must increase the amount of currency, i.e. the total currency in circulation under social institute framework is always expanding. When the increase of total currency doesn't go along with the increase of a country's total wealth (overall strength), inflation or deflation occurs. Currency has the most fluidity; it can be used to purchase other commodities with less fluidity, suchas real commodities or virtual commodities. Once the commodity is purchased with currency, it stops moving, but the currency keeps moving ahead. The currency transfers from the currency owner to the commodity owner. Currency becomes the value standard accepted by people. Since its separation from commodity, especially gold, currency has no value of storage. The idea that cash holding has opportunity cost becomes people's belief. Because currency is widely accepted by people, it has the absolute fluidity, and it moves toward the profitable direction. The taking-in and giving-out of currency by economic units are done in a flash. These currency, together with the excess of cash for the sake of adaptability preference, become the trade medium of all society. Having seepage feature, they move along the social institute framework and interact with it. Because the overall return of a country's currency expenditure is the summation of return of micro economic units'currency expenditure, it means GDP to a country.B. GDP dynamics modela. Conceptions(1) The definition of money circulation velocityMoney circulation velocity becomes larger when the return of money expenditure becomes larger and vice versa. The return of money expenditure is proportional to the money circulation velocity. The minor changes in return of money expenditure (GDP) will certainly lead to the changes in expenditure. Also, the minor changes of money expenditure will certainly lead to small changes in GDP. The proportionality coefficient is subject to many factors and is an institutional coefficient. Through the above analysis, we can get the mathematical expression of money circulation velocity as follows:In (1), V is money circulation velocity; R is money expenditure return (GDP); C is the expenditure on commodities and labors; I is expenditure on real capital goods; S is expenditure on stocks; B is expenditure on bonds; the differential terms mean that the money expenditure return in different investment areas or at different investment stages are different. In other words, the changes of money expenditure return will result in the changes of money trade volume in all directions. The denominators of differential terms mean the changes of investment volume of economic units. Their proportional coefficient K is the index of money permeability in social institutional framework. It is the permeation coefficient of money, a very important coefficient in money conduction institution. According to the above expression, when the differential term is 1, permeation coefficient is equal to the money circulation velocity. Because the differential term has no dimension, the permeation coefficient has the dimension of velocity, i.e. the variation of money in unit time. Due to the difference institutions in investment areas, the value of K is different. The money circulation velocity could be adjusted according to different value of K. Fisher's Equation of Exchange studies money in macro view while Cambridge equation studies in micro view. The following money quantity theorists, including Friedman, mainly continue to apply the micro research approaches of Cambridge equation. The money circulation velocity defined in this paper has apparently combined the macro variable—GDP and the micro variable—money expenditure together, and at the same time, combined money income GDP and money expenditure, money flux and stock, money quantity variation and money operational systems. From the above definition of money circulation velocity, we see that money movement is driven by the money expenditure return.(2) The elastic variation of cash holding balancesNow let's have a look at the situation in which the variation of cash holding is caused by the variation of money expenditure return. When the investor can't get back his prime cost by his investment, the cash holding balances will increase. When there's very little profit from investment, the cash holding balances will remain the same. Only when the money expenditure return is large enough can the cash holding balances begin to decrease. There is an active value of money expenditure return: when the return increases above the active value, the cash holding balances decrease; when it decreases to a certain level, it decreases very slowly, and even remains unchanged. When the money expenditure return is large enough, people will give up fluidity preference completely, and the cash holding balances will decrease to zero. This is the critical point of cash endogenous supply.In order to numerically estimate the above features, we have to make clear the correlation between the variation of money stock in social institutional framework (economic unit) and the corresponding variation of money expenditure return. Cash stock rate (elastic cash supply degree)βor cash stock coefficient (elastic release coefficient) A is applied to indicate the correlation. Cash stock rate shows the increasing amount of cash holding balances when the money expenditure return decreases by one unit (e.g. 1 RMB). Its dimension is [money unit]-1. It can be shown thus: A=βm (2)In expression (2), cash stock coefficient A indicates the giving-out (taking-in) volume of money from cash holding balances when the money expenditure return decreases (increases) by one unit. From the definition, we can see that cash stock coefficient has no dimension. The value of A differs according to different money operational institutions or different expenditure habits of economic units. This principle of the release and stock of cash is different from that of Tobin's cash inventory theory.(3) The continuous function of money movementAssuming that money is not destroyed, the cash inflow into economic units in unit time is Q1, and the outflow is Q2, and if Q1≠Q2, then the redundant cash is kept by the economic unit. This part will cause the cash holding variation△m, Q1-Q2=△m. If we assume that the inflow and outflow of money value remains the same, and the cash holding balances remains unchanged, then Q1-Q2=0 or Q1=Q2. In another word, the inflow of money is the same as the outflow.b. Basic differential equationThe money inflow into economic units duringΔt is: m·Δt (3)In the expression, m is money inflow into economic units in unit time.In the timeΔt, the outflow of money from economic units goes along four directions: Purchase of commodities and labor, real producing factors, deposit certificates and stock. First let's define the differential variable term of money movement in one of the directions. For the unit money used to purchase consumption of goods and labor, the minor changes of money result in the changes of money circulation velocity. Assuming that the consumed unit money is 1 money unit, then the differential variable term of money movement in the direction of consumption can be expressed by"". In the same way can we get the differential variable terms of the other three directions. The summation of four differential variable terms in four respective directions constitutes the difference of money inflow and outflow of economic units. The resource of money amount difference is from the changes of cash holding balances. The outflow of money can be expressed as: InΔt, the difference between money inflow and outflow of economic units is:The cash holding balances changes inΔt. The change in endogenous money can be derived by the definition of elastic supply degree of cash holding balances.This expression shows that the cash holding balances is an elastic storage quantity; it changes with money expenditure return (R). This relationship can be expressed by elastic supply degree of cash holding balances (β) or elastic release coefficient (A). Here, the cash holding balances is relevant to time. That is, the amount of cash holding balances will change under the influence of external factors, but there is a time lag. At the same time, its change shows the velocity of money amount change. The cash holding balances changes more quickly when the elasticity caused by external factors becomes greater. This will lead to impacts on all trade links. In this case, the external factors become important tools of monetary policies. The money amount difference should be equal to the money inflow and outflow of economic units, which is as follows: Substituting the above into the definition of money circulation velocity, we get:Expression (8) is the basic differential equation of unstable money movement when cash holding balances change. Assuming that the money institution is the same as the purchase of commodities, deposit, stock and real factors, i.e. Kc=KI=Ks=KB=K, we get:In order to facilitate the calculation, we introduce a parameter—money conductivity coefficient T. T=Km (10)In the expression, m is the cash holding balances; K is the permeation coefficient; money conductivity coefficient is the function of institutional framework, which differs with the institution of money movement. The money conductivity coefficient T and elastic release coefficient A are both important parameters for money movement. Substituting T , A into k,βof the equation (9), we get:Expression (11) is the basic differential equation of unstable money movement in the changing of cash holding balances. Actually, the elastic release amount of cash holding balances include both the variation caused by one unit change of money expenditure return plus the variation caused by the change of money value. As for the latter variation Friedman has defined it as practical cash holding balances to some extent, but he failed to give further analysis. When the money expenditure return in the right hand side of expression (11) is constant over time, , that is:Expression (12) is usually called the Laplace equation. It shows that the money is in a stable movement.The factors that lead to the variation of money amount in the unit are:①Outflow of domestic money results in the decrease of money amount in seepage field; the increase of money issue or the inflow of foreign money makes the money amount in seepage field increase. This kind of increase or decrease in money seepage field also takes the form of taking-in and giving-out between economic units, as well as the form of cash holding balances. We can use the following function to describe the variation of money amount of economic units in unit time: dm/dt = f(C,I,S,B;t) (13) The independent variable of this function is the purchase of consumption goods (C), purchase of real factors (I), purchase of stocks (S), purchase of bonds (B) and time (t). , The function takes a positive value when money inflows to the field.②Inflow money amount in upper stream of transaction links③Outflow money amount in lower stream of transaction links How factor②and③affects the money amount has been elaborated in detail. We only need to add expression (13) to the left part of (11), so that we can get the basic differential equation of money unstable movement in the seepage field where exogenous money joins in.In order to facilitate calculation, we makeα= TA , whereαis the conductivity coefficient of money expenditure return. These coefficients are related to monetary institutions. By this, equation (15) becomes:Expression (16) is the basic differential equation of money unstable movement of exogenous money. Strictly speaking, the money permeation coefficient K is different in different investment areas or directions. K is not a constant, but a function, so expression (16) is nonlinear equation. In order to solve the equation easily, different K at each areas should been seen as equal and a constant. The elastic supply degree of cash holding balances should also be seen as constant. Setting an initial value functionφ(C,I,S,B) and applying the Fourier transformation, then we get the solution of money expenditure return from the equation. (see appendix). The expression is as follows: III. DiscussionBy the description of money M0's movement in social institutional framework, we can abstract the economic unit's behavioral choice of cash holding balances. The economic unit's pursuit of money expenditure return is obviously regarded as a sensible natural process. Assuming that the money is not destroyed in circulation, they are either taken in, or given out, or kept as cash. The basic differential equation of money movement is derived which defines the money circulation velocity. It is subject to the heat conduction equation of liquids, so the money movement has the feature of fluids. Finally the initial value borderline is given; the mathematical expression of money expenditure return, i.e. GDP is obtained by which we solve the equation. Transforming expression (18) by polar coordinates to an integral gives a maximum value of 1. We can apply the extreme value estimate to the mathematical expression (17) of GDP.According to the law of maximum mold estimation of initial value solution: R (C , I , S , B; t )∈C2,2,2,2;1(Q )∩C (Q|-)This is the solution of initial value problems. Then we can apply the following estimate to expression (17): SupQ|R(C ,I,S,B;t)|≤t.SupQ|α. f(C,I,S,B;t)|+SupQ|φ(C,I,S,B) (19)From Expression (19),, the extreme value of a country's money expenditure return during a certain period is less than or equal to the extreme value at the beginning added to the extreme value derived by exogenous money amount multiplied by institutional coefficient. This value shows that given other factors, the growth of GDP is only related to the increase of exogenous money, endogenous money and institutions. This is becauseαreflects the institutional state in the areas of cash residue, elastic supply and money expenditure return. The"less than or equal to"symbol in this expression indicates that the equation could only stand when the economy is in best condition. In real economic operation, the"less than"is always applied because there is always frictions in institutions so that money can't reach the best circulation condition. If the economic operation condition is a constant at the beginning, the growth of GDP is only proportional to the changes of exogenous and endogenous money. The proportional coefficient is institutional factors and time. If we set the time to 1 year, then the expression of GDP is a straight line with the extreme value of GDP at the beginning as intercept, and with the institutional factors, i.e. money expenditure return conductivity coefficientαas slope. In other words, the expression of GDP is a linear function.A. The endogenous changes of moneyThe transaction demand causes the changes of cash holding balances for fluidity preference in households, enterprises, banks and governments fountain. And the fountain and collection are formed.[ARE THESE THE CORRECT TERMS– FOUNTAIN ??] When the money expenditure return increases, the cash holding balances in economic units decreases, and fountain is formed; otherwise, collection is formed. This change in cash holding balances will lead to the increase or decrease of moving money in the seepage field, thus which will cause the disturbance of the money seepage field. The disturbance will increase, indicating that the economic operation system is in an unstable condition; otherwise with time, it is in a stable condition. Since people are always endeavoring to create more and more wealth-goods to satisfy their needs, we have to increase the amount of money issue because the overall amount of GDP is increasing even although the unit money expenditure return remains unchanged in micro view. The standard of money expenditure return is the interest rate issued by banks, just as with the consumption and money standards. Therefore, the interest rate becomes the vital factor which influences the variation of cash holding balances of economic units.In addition, credit system, financial infrastructure, innovation of financial instruments, market trade institutions and cultural traditions are all secondary factors that influence the variation of cash holding balances.B. The exogenous changes of moneyThe exogenous changes of money refers to the money changes in circulation, which is caused by the entry or exit of either foreign or domestic money, or the increased money issue in order to achieve the nation's strategy. In view that money movement is subject to the heat conduction equation, the money movement has the features of fluids. We can only control the borderline of fluid movement. We can not control the inside of fluids. The principle of increasing money issue is to keep the money value unchanged, i.e. keep the purchasing power unchanged. Therefore, the increase of money issue must go along with the increase of wealth-goods. We must at least make"M0V"no smaller than the increase of wealth-goods, so that we can stimulate people to create more wealth-goods.Because money movement has the features of fluids, it has energy. Just as with the three kinds of energy in fluid movement, money movement energy also includes three parts: the energy that relates to velocity; the potential energy that relates to money expenditure return (GDP), and the money value energy that relates to purchasing power. It is no doubt that a country's monetary fiscal policy must seek to increase the energy of money movement because if the energy of money movement is increased, the external expansion ability of the country is increased so that it is helpful in achieving the nation's strategies. The following measures in three aspects should be taken to increase the energy of money movement. Firstly, establish the institutions that increase the money movement velocity. Secondly, increase the GDP growth rate, while keeping the growth rate steady. Thirdly, the value of money is increased by the increase of wealth-goods. The increase of exogenous money should be help to increase the energy of money movement. C. The conductivity coefficient of money expenditure return:αIn the deduction of the model,α= T/A =(Km)/(βm)=K/β. In the expression, K is the money permeation coefficient that relates to money circulation velocity, or in other words, relates to institutions. Therefore it is the institutional coefficient that is relevant to the transaction activities in which money is used to purchase wealth-goods in the four directions.Βis the institutional coefficient that is relevant to the changes of cash holding balances. We can then see thatαis the characteristic factor variable that involves the whole money movement institution. In the deduction process of the model, we assume that the money permeation coefficient is equal in the four directions, but this doesn't influence the model's interpretation of economic operation features.In expression (19), as the slope of GDP straight line equation,αcontrol the growth track of GDP. In other words, changes of monetary financial policy (institution) will result in the changes of nonlinearity in GDP growth and cause chaos to economy. In this sense, it is of vital importance to assure the stability of a country's monetary fiscal policy to achieve the stable and efficient operation of economy. IV. Empirical TestWhether the theoretical model can stand the proof shows whether the theory has power. The author chooses the value of GDP and the cash in circulation (M0) to apply regression analysis to the model, so that we can see its fitting degree. The following is the data from China Statistics Year Book.Source: China statistics year book of 2004Suppose that t equals one in expression (19). The expr...
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