Fisher's quantity theory is best explained with the help of his famous equation of exchange.

MV_{T}=P_{T}T (12.1)

where the subscript T is added to V and P to emphasise that they relate to total transactions. Each side of the equation gives the money value of total transactions during a period. Let us see how. First consider the right-hand side of the equation. In the case of a single (say ith) transaction, with its price p_{i} and quantity t_{i}, its money value will be given by P_{i}t_{i}. When money value of ail such transactions, whether of goods, services, or assets, etc. are added up, we get get ∑_{i}p_{i}t_{i}

Taking P_{T} as a suitably chosen average of all prices P_{i} and T as a suitably chosen aggregate of all quantities transacted t_{i}, we have

P_{T}T= ∑ /i P_{i} t_{i} (12.2)

Now consider the left-hand side of equation MV_{T}=P_{T}T (12.1). In it M is the total quantity of money in the economy and V_{T} is its transactions velocity, that is, the average number of times a unit of money changes hands to effectuate transactions during the period chosen. Then, MV_{T} will also give the money value of total transactions during the same period. Since ex post both MV_{T} and P_{T}T measure the same total (money value of transactions during a period), the two must be equal to each other. Hence equation MV_{T}=P_{T}T (12.1). That is why it is also called equation of exchange.

Thus interpreted, equation MV_{T}=P_{T}T (12.1) is an identity. Since ex post it must always be true, it is also a truism. Why is it then called a theory — a theory which says that a change in the quantity of money will lead to an equi-proportionate change in P in the same direction?

Before discussing the answer of the QTM, a general point may be made. Equation MV_{T}=P_{T}T (12.1) is one equation in four unknowns (or variables). Therefore, it can be used to solve for the value of only one of them in terms of the other three. That is given the values of any three variables, the value of the fourth one has to be such as to satisfy the equation MV_{T}=P_{T}T (12.1). What the QTM does specifically is to assume that T and V_{T} are constants, or at least autonomous of changes in M, that changes in M are autonomous of the other three (P, V_{T}, and T), and that consequently, changes in M lead to equi-proportionate changes in P.

Since it highlights the relation between M and P and makes changes in the former the (major) cause of changes in P, it becomes 'a quantity of money theory of P. Popularly it is called the QTM.

**This is explained more fully below:**

We begin with the assumptions of the QTM with respect to individual factors (T.M.V_{T}) and P_{T}) assembled in the equation.

**Transactions (T):**

In the QTM it is assumed that the physical volume of transactions (T) is determined by the basic physical and operational characteristics of the system, such as the real resources available to the economy, the efficiency with which they are used, the degree of business integration of the economy (which determines the number of transactions involved in the production and sale of final goods).

More important, "all quantity theorists, at least since Hume, have recognized that changes in the stock of money may have transitional effects on T. However, they have generally regarded the average level of T and long-run changes in T as largely independent of the quantity of money although not of the existence of a money economy". In addition, it is assumed that changes in V_{T} and P_{T }do not influence T (except temporarily). Thus, the demand-side influence on T is neglected completely.

**Money (M):**

It is assumed that the factors determining the stock of M depend critically on the monetary system and are largely independent of the forces determining T.

**Velocity of Circulation (V**_{T}):

_{T}):

The QTM is often associated with the assumption of a constant V—that V is something of a natural constant. This is not fully correct. No doubt, the transactions approach emphasises payment practices, such as the frequency with which people are paid, the irregularity of receipts and payments, as its key determinant. But Fisher and earlier quantity theorists did explicitly recognize that velocity would also be affected by, among other things, the rate of interest and also the rate of change of prices.

They recognized that both high rates of interest and rapidly rising prices would induce people to economies on money balances and so tend to raise velocity and that low rates of interest and falling prices would have the opposite effect.

However, all this was not woven systematically into a complete macro model. It was also assumed that payment practices, though responsive to cost considerations, were rather slow to change. Therefore, it was thought to be a good first approximation to assume that V was almost a constant.

**Prices (P**_{T}):

_{T}):

P_{T }refers to the average price of market transactions of all kinds, whether in currently-produced final goods or services or intermediates, or old goods, or transactions of a purely financial nature. In the QTM, P_{T} is treated as the dependent variable. Assuming T and V_{T} to remain unchanged, or rather autonomous of changes in M, it makes P_{T} alone as the factor that absorbs all changes in M. That is equation MV_{T}=P_{T}T (12.1) can be used to solve for the value of P_{T} which will make the two sides of this equation equal. This gives

P_{T }= MV_{T}/T (12.3)

In the rigid version of the QTM presented above, P_{T} is seen to be a proportional function of M (given V_{T} and T) a doubling of the quantity of M will lead to the doubling of P. Also, since changes in M are assumed to be autonomous of P_{T} the former are made the cause of changes in the latter. This sums up the theoretical content of the QTM (transactions approach).

**Another Version****:**

Following Fisher, it is customary to subdivide the left-hand side of equation MV_{T}=P_{T}T (12.1) into two categories of payments those effected by the transfer of currency (including coins) and those effected by the transfer of demand deposits. (Recall the commonly-accepted definition of money as the sum of currency and demand deposits.).

One reason for the emphasis on this kind of division was the belief that the velocities of circulation of the two kinds of money were different. The other reason was the ready availability of data on bank clearings and so on the turnover or velocity of bank deposits.

Redefining M as only currency and V_{T} as its velocity, M as the volume of demand deposits and M for their velocity, equation MV_{T}=P_{T}T (12.1) can be recast as

MV_{T} + M'V_{T}= P_{T}T. (12.4)

This is not important enough to be pursued further.

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