The equation of exchange is simply the sum of the equations involved in all individual exchanges in a year. In each sale and purchase, the money and goods exchanged are _ipso facto_ equivalent; for instance, the money paid for sugar is equivalent to the sugar bought. And in the grand total of all exchanges for a year, the total money paid is equal in value to the total value of the goods bought. The equation thus has a money side and a goods side. The money side is the total money paid, and may be considered as the product of the quant.i.ty of money multiplied by its rapidity of circulation. The goods side is made up of the products of quant.i.ties of goods exchanged multiplied by their respective prices.
The important magnitude, called the velocity of circulation, or rapidity of turnover, is simply the quotient obtained by dividing the total money payments for goods in the course of a year by the average amount of money in circulation by which those payments are effected. This velocity of circulation for an entire community is a sort of average of the rates of turnover of money for different persons. Each person has his own rate of turnover which he can readily calculate by dividing the amount of money he expends per year by the average amount he carries.
Let us begin with the money side. If the number of dollars in a country is 5,000,000, and their velocity of circulation is twenty times per year, then the total amount of money changing hands (for goods) per year is 5,000,000 times twenty, or $100,000,000. This is the _money_ side of the equation of exchange.
Since the money side of the equation is $100,000,000, the goods side must be the same. For if $100,000,000 has been spent for goods in the course of the year, then $100,000,000 worth of goods must have been sold in that year. In order to avoid the necessity of writing out the quant.i.ties and prices of the innumerable varieties of goods which are actually exchanged, let us a.s.sume for the present that there are only three kinds of goods,--bread, coal, and cloth; and that the sales are:
200,000,000 loaves of bread at $ .10 a loaf, 10,000,000 tons of coal at 5.00 a ton, and 30,000,000 yards of cloth at 1.00 a yard.
The value of these transactions is evidently $100,000,000, _i. e._, $20,000,000 worth of bread plus $50,000,000 worth of coal plus $30,000,000 worth of cloth. The equation of exchange therefore (remember that the money side consisted of $5,000,000 exchanged 20 times) is as follows:
$5,000,000 20 times a year = 200,000,000 loaves $ .10 a loaf + 10,000,000 tons 5.00 a ton + 30,000,000 yards 1.00 a yard
This equation contains on the money side two magnitudes, viz. (1) the quant.i.ty of money and (2) its velocity of circulation; and on the goods side two _groups_ of magnitudes in two columns, viz. (1) the quant.i.ties of goods exchanged (loaves, tons, yards), and (2) the prices of these goods. The equation shows that these four sets of magnitudes are mutually related. Because this equation must be fulfilled, the prices must bear a relation to the three other sets of magnitudes--quant.i.ty of money, rapidity of circulation, and quant.i.ties of goods exchanged.
Consequently, these prices must, as a whole, vary proportionally with the quant.i.ty of money and with its velocity of circulation, and inversely with the quant.i.ties of goods exchanged.
Suppose, for instance, that the quant.i.ty of money were doubled, while its velocity of circulation and the quant.i.ties of goods exchanged remained the same. Then it would be quite impossible for prices to remain unchanged. The money side would now be $10,000,000 20 times a year or $200,000,000; whereas, if prices should not change, the goods would remain $100,000,000, and the equation would be violated. Since exchanges, individually and collectively, always involve an equivalent _quid pro quo_, the two sides _must_ be equal. Not only must purchases and sales be equal in amount--since every article bought by one person is necessarily sold by another--but the total value of goods sold must equal the total amount of money exchanged. Therefore, under the given conditions, prices must change in such a way as to raise the goods side from $100,000,000 to $200,000,000. This doubling may be accomplished by an even or uneven rise in prices but some sort of _a rise of prices there must be_. If the prices rise evenly, they will evidently all be exactly doubled.... If the prices rise unevenly, the doubling must evidently be brought about by compensation; if some prices rise by less than double, others must rise by enough more than double to exactly compensate.
But whether all prices increase uniformly, each being exactly doubled, or some prices increase more and some less (so as still to double the total money value of the goods purchased), the prices _are_ doubled _on the average_.... From the mere fact, therefore, that the money spent for goods must equal the quant.i.ties of those goods multiplied by their prices, it follows that the level of prices must rise or fall according to changes in the quant.i.ty of money, _unless_ there are changes in its velocity of circulation or in the quant.i.ties of goods exchanged.
If changes in the quant.i.ty of money affect prices, so will changes in the other factors--quant.i.ties of goods and velocity of circulation--affect prices, and in a very similar manner. Thus a doubling in the velocity of circulation of money will double the level of prices, provided the quant.i.ty of money in circulation and the quant.i.ties of goods exchanged for money remain as before....
Again, a doubling in the quant.i.ties of goods exchanged will not double, but halve, the height of the price level, _provided_ the quant.i.ty of money and its velocity of circulation remain the same....
Finally, if there is a simultaneous change in two or all of the three influences, _i. e._, quant.i.ty of money, velocity of circulation, and quant.i.ties of goods exchanged, the price level will be a compound or resultant of these various influences. If, for example, the quant.i.ty of money is doubled, and its velocity of circulation is halved, while the quant.i.ty of goods exchanged remains constant, the price level will be undisturbed. Likewise, it will be undisturbed if the quant.i.ty of money is doubled and the quant.i.ty of goods is doubled, while the velocity of circulation remains the same. To double the quant.i.ty of money, therefore, is not always to double prices. We must distinctly recognize that the quant.i.ty of money is only one of three factors, all equally important in determining the price level....
We now come to the strict algebraic statement of the equation of exchange.... Let us denote the total circulation of money, _i. e._, the amount of money expended for goods in a given community during a given year, by _E_ (expenditure); and the average amount of money in circulation in the community during the year by _M_ (money). _M_ will be the simple arithmetical average of the amounts of money existing at successive instants separated from each other by equal intervals of time indefinitely small. If we divide the year"s expenditures, _E_, by the average amount of money, _M_, we shall obtain what is called the average rate of turnover of money in its exchange for goods, _E_/_M_ that is, the velocity of circulation of money. This velocity may be denoted by _V_, so that _E_/_M_ = _V_; then _E_ may be expressed as _MV_. In words: the total circulation of money in the sense of money expended is equal to the total money in circulation multiplied by its velocity of circulation or turnover. _E_ or _MV_, therefore, expresses the money side of the equation of exchange. Turning to the goods side of the equation, we have to deal with the prices of goods exchanged and quant.i.ties of goods exchanged. The average price of sale of any particular good, such as bread, purchased in the given community during the given year, may be represented by _p_ (price); and the total quant.i.ty of it purchased, by _Q_ (quant.i.ty); likewise the average price of another good (say coal) may be represented by _p"_ and the total quant.i.ty of it exchanged, by _Q"_; the average price and the total quant.i.ty of a third good (say cloth) may be represented by _p""_ and _Q""_ respectively; and so on, for all other goods exchanged, however numerous. The equation of exchange may evidently be expressed as follows:
_MV_ = _pQ_ + _p"Q"_ + _p""Q""_ + etc.
The right-hand side of this equation is the sum of terms of the form _pQ_--a price multiplied by a quant.i.ty bought. It is customary in mathematics to abbreviate such a sum of terms (all of which are of the same form) by using "Sigma" as a symbol of summation. This symbol does not signify a _magnitude_ as do the symbols _M, V, p, Q_, etc. It signifies merely the _operation_ of addition and should be read "the sum of terms of the following type." The equation of exchange may therefore be written:
_MV_ = Sigma_pQ_.
That is, the magnitudes _E_, _M_, _V_, the _p_"s and the _Q_"s relate to the _entire_ community and an _entire_ year; but they are based on and related to corresponding magnitudes for the individual persons of which the community is composed and for the individual moments of time of which the year is composed.
The algebraic derivation of this equation is, of course, essentially the same as the arithmetical derivation previously given. It consists simply _in adding together the equations for all individual purchases within the community during the year_....
[We are now] ... prepared for the inclusion of bank deposits or circulating credit in the equation of exchange. We shall still use _M_ to express the quant.i.ty of actual money, and _V_ to express the velocity of its circulation.[46] Similarly, we shall now use _M"_ to express the total deposits subject to transfer by check; and _V"_ to express the average velocity of circulation. The total value of purchases in a year is therefore no longer to be measured by _MV_, but by _MV_ + _M"V""_.
The equation of exchange, therefore, becomes:
_MV_ + _M"V"_ = Sigma_pQ_ = _PT_[47]....
With the extension of the equation of monetary circulation to include deposit circulation, the influence exerted by the quant.i.ty of money on general prices becomes less direct; and the process of tracing this influence becomes more difficult and complicated. It has even been argued that this interposition of circulating credit breaks whatever connection there may be between prices and the quant.i.ty of money.[48]
This would be true if circulating credit were independent of money. But the fact is that the quant.i.ty of circulating credit, _M"_, tends to hold a definite relation to _M_, the quant.i.ty of money in circulation; that is, deposits are normally a more or less definite multiple of money.
Two facts normally give deposits a more or less definite ratio to money.
The first ... [is] that bank reserves are kept in a more or less definite ratio to bank deposits. The second is that individuals, firms, and corporations preserve more or less definite ratios between their cash transactions and their check transactions, and also between their money and deposit balances.[49] These ratios are determined by motives of individual convenience and habit. In general, business firms use money for wage payments, and for small miscellaneous transactions included under the term "petty cash"; while for settlements with each other they usually prefer checks. These preferences are so strong that we could not imagine them overridden except temporarily and to a small degree. A business firm would hardly pay car fares with checks and liquidate its large liabilities with cash. Each person strikes an equilibrium between his use of the two methods of payment, and does not greatly disturb it except for short periods of time. He keeps his stock of money or his bank balance in constant adjustment to the payments he makes in money or by check. Whenever his stock of money becomes relatively small and his bank balance relatively large, he cashes a check. In the opposite event, he deposits cash. In this way he is constantly converting one of the two media of exchange into the other. A private individual usually feeds his purse from his bank account; a retail commercial firm usually feeds its bank account from its till. The bank acts as intermediary for both.
In a given community the quant.i.tative relation of deposit currency to money is determined by several considerations of convenience. In the first place, the more highly developed the business of a community, the more prevalent the use of checks. Where business is conducted on a large scale, merchants habitually transact their larger operations with each other by means of checks, and their smaller ones by means of cash.
Again, the more concentrated the population, the more prevalent the use of checks. In cities it is more convenient both for the payer and the payee to make large payments by check; whereas, in the country, trips to a bank are too expensive in time and effort to be convenient, and therefore more money is used in proportion to the amount of business done. Again, the wealthier the members of the community, the more largely will they use checks. Laborers seldom use them; but capitalists, professional and salaried men use them habitually, for personal as well as business transactions.
There is, then, a relation of convenience and custom between check and cash circulation, and a more or less stable ratio between the deposit balance of the average man or corporation and the stock of money kept in pocket or till. This fact, as applied to the country as a whole, means that by convenience a rough ratio is fixed between _M_ and _M"_. If that ratio is disturbed temporarily, there will come into play a tendency to restore it. Individuals will deposit surplus cash, or they will cash surplus deposits.
Hence, both money in circulation ... and money in reserve ... tend to keep in a fixed ratio to deposits. It follows that the two must be in a fixed ratio to each other.
It further follows that any change in _M_, the quant.i.ty of money in circulation, requiring as it normally does a proportional change in _M"_, the volume of bank deposits subject to check, will result in an exactly proportional change in the general level of prices except, of course, so far as this effect be interfered with by concomitant changes in the _V_"s or the _Q_"s. The truth of this proposition is evident from the equation _MV_ + _M"V"_ = Sigma_pQ_; for if, say, _M_ and _M"_ are doubled, while _V_ and _V"_ remain the same, the left side of the equation is doubled and therefore the right side must be doubled also.
But if the _Q_"s remain unchanged, then evidently all the _p_"s must be doubled, or else if some are less than doubled, others must be enough more than doubled to compensate....
The factors in the equation of exchange are ... continually seeking normal adjustment. A ship in a calm sea will "pitch" only a few times before coming to rest, but in a high sea, the pitching never ceases.
While continually seeking equilibrium, the ship continually encounters causes which accentuate the oscillation. The factors seeking mutual adjustment are money in circulation, deposits, their velocities, the _Q_"s and the _p_"s. These magnitudes must always be linked together by the equation _MV_ + _M"V"_ = Sigma_pQ_. This represents the mechanism of exchange. But in order to conform to such a relation the displacement of any one part of the mechanism spreads its effects during the transition periods [_i.e._, periods of rising or falling prices]
over all parts. Since periods of transition are the rule and those of equilibrium the exception, the mechanism of exchange is almost always in a dynamic rather than a static condition....[50]
[Ill.u.s.tration]
[51]It is interesting to make a quant.i.tative comparison of the various magnitudes with the increase in the quant.i.ty of money as the most important factor in raising the price level. While it is true, as shown by the diagram, that the volume of deposits subject to check has increased greatly, the major part of the increase has to be ascribed to the increase in the quant.i.ty of money. Only so far as the volume of deposits subject to check has increased relatively to the money in circulation, can the increase of deposits be regarded as an independent cause of the rise in prices. We have thus to consider the relative importance of the five causes affecting prices:
1. The quant.i.ty of money in circulation (M).
2. The volume of bank deposits subject to check considered relatively to money (M"/M).
3. The velocity of the former (V").
4. The velocity of the latter (V).
5. The volume of trade (T).
We may best compare the relative importance of these five magnitudes by answering the question: What would the result have been had any one of these magnitudes remained unchanged, a.s.suming that the other four changed in the same manner that they actually did change. We find (1) that if the money in circulation, M, had not changed, between the years 1896 and 1909, for example, the price level of 1909 would have been 45 per cent. lower than it actually was; (2) that if M"/M, the relative deposits, had not changed, during the same period the price level in 1909 would have been 23 per cent. lower than it actually was; (3) if the velocity of circulation of money, V, had not changed, the price level for 1909 would have been 1 per cent. lower; (4) if the velocity of circulation of deposits, V", had not changed, the price level in 1909 would have been 28 per cent. lower; (5) if T had not changed, the price level in 1909 would have been 106 per cent. _higher_.
Thus the changes in the first four factors have tended to raise prices, while the change in T has tended to lower prices. The relative importance of the four price-raising causes may be stated in terms of the per cent. already given which represents how much lower prices would have been except for each of these causes separately considered.
According to this test we find the relative importance of the four price-raising factors to be as follows:
The importance of V is represented by 1,
The importance of M"/M is represented by 23,
The importance of V is represented by 28,
The importance of M is represented by 45.
That is, the increase in the quant.i.ty of money had an importance nearly double that of any other one price-raising factor, during the period mentioned.
INDIRECT INFLUENCES ON PURCHASING POWER[52]
Thus far we have considered the level of prices as affected by the volume of trade, by the velocities of circulation of money and of deposits, and by the quant.i.ties of money and of deposits. These are the only influences which can _directly_ affect the level of prices. Any other influences on prices must act through these five. There are myriads of such influences (outside of the equation of exchange) that affect prices through these five. It is our purpose ... to note the chief among them....
We shall first consider the outside influences that affect the volume of trade and, through it, the price level. The conditions which determine the extent of trade are numerous and technical. The most important may be cla.s.sified as follows: