Monetary Economics/The Quantity Theories

History
This theory was described comprehensively by Irving Fisher (1911), in the book The Purchasing Power of Money. It is the classical view of how money is used in the economy, and what variables it affects. The quantity theory of money was initially known as the equation of exchanged.

Fisher's Model
Consider a closed economy. In every transaction, the value of sales and the value of purchases must match. Therefore, even when scaled up to the aggregate level, the value of payments must equal the value of receipts.

$$P_t \equiv R$$

But $$P_t$$, the value of payments, must be equal to the number of transactions multiplied by the average price they take place at. $$R$$, the value of receipts, must be equal to the amount of money in the economy multipled by the average number of times it circulates (or, the velocity of money).

This gives us an identity:

$$ PT \equiv M_sV_T $$

where $$P$$ is simply the price level and $$T$$ is the volume of transactions. $$M_s$$ is the supply of money in the economy, and $$V_T$$ is termed the velocity of transaction.

Inflation
Consider now the behaviour of these variables in the real world. The money supply is of course exogenously determined (by the government / central bank). We can assume that in equilibrium the market clears and therefore the supply of money $$M_s$$ is equal to the demand for money $$M_d$$.

Fisher further assumed that the volume of transactions $$T$$ would be relatively stable at long run equilibrium, and that the velocity of money $$V_T$$ would also be invariant in the short run. This would then give us the following equation:

$$M_dV_T = PT \,$$

$$P = {M_dV_T \over T}$$

It can be seen that any change to the price level is wholly induced by changes to the exogenous variables. If we consider $$V_T$$ and $$T$$ invariant over the short run, the rate of change in $$P$$ is entirely due to the rate of change in $$M_d$$. That is, the rate of inflation in the economy is entirely due to changes in the money supply. The equilibriating mechanism is the change in money demand.

The Neutrality of Money
Consider the basic equation of the model.

$$M_dV_T = PT \,$$

We then have:

$${M_d \over P} = {T \over V_T}$$

If the Fisherian quantity theory is correct, then any change in $$M_d$$ would lead to a corresponding change in $$P$$, while the real variables, $$T$$ and $$V_T$$, remain unchanged. This is known as the neutrality of money, a condition whereby changes to the money supply affect only nominal variables. A related, and harder condition to satisfy, is the superneutrality of money, where changes in the rate of growth of the nominal money supply influence only the rate of inflation, and the rate of depreciation of the currency in an open economy. There is no effect on any other variable.

History
This preliminary version of the quantity theory was developed by Marshall and Pigou (1917) in order to extend the analysis to consider short run fluctuations in the velocity of money. In Fisherian quantity theory the velocity of money was determined exogenously.

A change in viewpoint
Under the Fisherian analysis an individual holds as much money as he needs in order to carry out transactions. Marshall and Pigou approached the problem slightly differently. Their version of the quantity theory determines how much money an individual chooses to hold in order to carry out transactions.

The explicit assumptions here are that the individual cannot hold more money than his stock of wealth, that he makes a choice between money, which does not pay interest income, and bonds, which do. The demand for money function therefore varies with the interest rate, the level of wealth, and the volume of transactions that said individual wishes to carry out.

The model proposed by Marshall and Pigou was simplified by assuming that the volume of transactions would be stable in the short run. This allowed the aggregate real money demand function to be equated to some proportion of the level of real income in the economy. We thus have:

$${M_d \over P} = kY$$

Rearranging:

$$M_d = kPY \,$$

And:

$$\left ( M_d \right ) \left ( \frac{1}{k} \right ) = PY$$

Here, the constant $$k^{-1}$$ is the income velocity of money, as opposed to $$V_T \,$$ in the Fisherian analysis, which was the transactions velocity of money.

Keynesian analysis - the introduction of the interest rate
Keynes' contribution to the above analysis focused on defining more clearly the motives economic agents had in holding money. He defined 3 motives for holding money - the transactions motive (where agents hold money in order to make planned payments), the precautionary motive (where agents hold money against the unforeseen need to make payments) and the speculative motive (where agents make the decision to hold money according to the opportunity cost of doing so).

The speculative motive can be explained more clearly by considering the asset allocation decision between money and bonds. We defined bonds earlier as any asset that pays an income. The rate of this income is, quite clearly, the interest rate. If the interest rate is expected to fall, the price of bonds will rise, and if the interest rate is expected to rise, the price of bonds will fall.

Keynes extended the analysis by assuming that there is a range of interest rates that is considered to be the mean range, and that the interest rate would be mean-reverting towards it.