21 The Quantity Equation

Already the French political theorist Bodin (1576) recognized basic ideas of the later quantity theory. The first workable formulation of the quantity theory comes from the English philosopher Locke (1689), where he introduced the term of monetary velocity. One of the most famous representative of the quantity theory of money was the U.S. economist Milton Friedman (1912-2006). In its essential expression the quantity theory asserts the relationship

    (21.1).

The significance is that the product of the circulating money supplytimes the velocity of circulationequals the product of the price leveltimes the trading volume, which is the frequency of transactions of real goods. The trading volume times the price level of the gross domestic product equals the GDP.

    (21.2)

A fundamental problem is, however, that no doubt all four function depends on time

    (21.3).

However this temporal relationship remains unknown, because of a conditional equation alone, the four solution functions are impossible to be clearly determined. Therefore, this well known connection is usually used only as a rule of thumb. After all, holding two of the variables constant, one can imagine the influence of a change in the size of the third to the fourth variable. For local situations, which means if the formula is used for only a short timeline, and so the variation of other sizes may be realistically considered negligible, then the quantity equation in this form can be already very helpful.

Now this equation is also a typical balance equation, because it is merely a mathematical formulation of the principle of trade "money for goods." In a higher valid theory, the quantity equation must be included and in that theory the time dependencies of the individual functions in principle should also be determinable analytically.

Since it is usually treated only as a rule of thumb, it goes into the classical literature with a more generous treatment of the units. Sometimes we equate H with Y, which is not true, and P is often identified with inflation, which thus also is not right. In addition, M is identified usually with one of the three monetary aggregates, and finally the velocity V is set to any size, e.g. between 0 and 1, for non-existent or inaccurate definition of its unit.

First, we need to ensure a workable definition of the four factors. While the definition problems are relatively clear for H, P and Y, the problem is more difficult for what aggregate the money M represents, and which V should be defined correctly, which is supposed to not be seen so simply.

V is the speed for revolving the liquid money in the economy, and we would suggest a similar motion of velocity like m/sec in mechanics, so the unit is actually currency/year. The M funds should be, at least it is often thought so in literature, just be that liquid money supply aggregate. But then one gets

as a badly defined unit for the left hand side of the quantity equation. On the right hand side one often takes the price level in currency and the volume of trade as an unspecified number of the unit 1. This results on the right side at the unit

,

what is not properly absorbed. One can twist and turn in different ways, but in each case either the units on both sides do not match, or the unit of a single size does not make sense. Units should be in fact both sides identical and due to Y=HP have the unit currency/year:

    (21.4)

Since money is measured in Cur the unit equation above is necessarily for a solution. Here the unit Buy is for a purchase or a number of purchases. Now the units on both sides agree. H is the number of purchases per year and the price level is the average price per purchase. So is the right side

    (21.5).

The left side is not as fast to make clear. For this it needs some considerations. The product (MV) should of course be the annual money supply in GDP. Classically one assumes as an approximation one of the money aggregatestowhich have in fact a different degree of determination (cash and short-term investment pays off as distinct from long-term frozen money or assets). But these are actually only approximations, since these money aggregates are present in the balance certainly, but whether they actually run around and be used for purchases is by no means certain. But since we need a complete balance equation for the overall economy, we must proceed differently.

For not the aggregatesare to be set for M, but the total capital stock K as a whole. Multiplied by the money velocity V, we get the actual amount of money in circulation that is

    (21.6)

which has now the correct unit

    (21.7).

The now well defined continuity equation is now written in expanded notation:

    (21.8).

Now the balance is alright because the circulating money (wages and revenues, but also duties and taxes, or withdrawals or loans from the Bank for investment and consumption) is of course the GDP Y as a whole. The measured monetary aggregatesare only a subset of the capital stock and just provide an approximation of.

As an example we can now compute the balance equation for the the year 2008 in the FRG: the total capital stock K was then about 8000 billion Euros [€]. V was the proportion of this as available money, here about 0.3/year = 30% [1/y]. H is the volume of trade, i.e. the number of purchases in 2008, which can by now only be roughly estimated: With the number of inhabitants in the FRG of about 83 millions and an average of one purchase a day, we assume about 83 million times 365 = 30 billion purchases per year [Buy/y].The price level is then the GDP divided by the number of purchases, so P can then be calculated with a GDP at that time of about 2400 billion € to 80 € per purchase [€/Buy]. As is easily seen, then the equation is balanced:

     

    (21.9)

The monetary velocity estimated here is also obtained from the real numbers of the Bundesbank, so by the monetary aggregateor the total credit amount into the GDP. These statistical numbers are only an approximation, but a pretty good one. Therefore we compute for the past year 2007 now specifically: In December 2007, the GDP € was 2975.7 billion and the total capital stock was € 7625.7 billion. The monetary aggregatewas communicated to 2187.8 billion € and the amount of outstanding loans to domestic non-banks in the FRG amounted to 2975.7 billion €.
Thus we get froma velocity of and by the loans to domestic non-banks we get, which are both approximatly by the nature of the underlying data. The population in 2007 was 73.941 million inhabitants. Multiplied by 365 days gives 26.988 billion purchases that year. So is calculated the price level in this specific example to about

    (21.10)

in the national economic average. The problem of the quantity equation, however, remains that there is still an under-determined equation system. Indeed, the calculation of the exact numbers are lacking further defining equations.