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### Fisher’s quantity theory of money: the cash transactions approach

The quantity theory of money states that the quantity of money is the main determinant of the price level or the value of money. Any change in the quantity of money produces an exactly proportionate change in the price level. In the words of Irving Fisher, “Other things remaining unchanged, as the quantity of money in circulation increases, the price level also increases in direct proportion and the value of money decreases and vice versa.” If the quantity of money is doubled, the price level will also double and the value of money will be one half. On the other hand, if the quantity of money is reduced by one half, the price level will also be reduced by one half and the value of money will be twice.

Fisher has explained his theory in terms of his equation of exchange:

PT = MV + M’ V’

where P = price level, or 1/P = the value of money;

M = the total quantity of legal tender money;

V = the velocity of circulation of M;

M’ = the total quantity of credit money;

V’ = the velocity of circulation of M’;

T = the total amount of goods and services exchanged for money or transactions performed by money.

This equation equates the demand for money (PT ) to supply of money (MV=M’V’). The total volume of transactions multiplied by the price level (PT) represent the demand for money. According to Fisher, PT is ΣPQ. In other words, price level (P) multiplied by quantity bought (Q) by the community (Σ) gives the total demand for money. This equals the total supply of money in the community consisting of the quantity of actual money M and its velocity of circulation V plus the total quantity of credit money M’ and its velocity of circulation V’. Thus the total value of purchases (PT) in a year is measured by MV+M’V’. Thus the equation of exchange is PT=MV+M’V’. In order to find out the effect of the quantity of money on the price level or the value of money, we write the equation as

Fisher points out that the price level (P) varies directly as the quantity of money (M+M’ ), provided the volume of trade (T) and velocity of circulation (V, V’ ) remain unchanged. The truth of this proposition is evident from the fact that if M and M’ are doubled, while V, V’ and T remain constant, P is also doubled, but the value of money (1/P) is reduced to half.

Fisher’s quantity theory of money is explained with the help of Figure 1 (A) and (B). Panel A of the figure shows the effect of changes in the quantity of money on the price level. To begin with, when the quantity of money is M2, the price level is P2. When the quantity of money is doubled to M, the price level is also doubled to P. Further, when the quantity of money is increased four-fold to M1, the price level also increases by four times to P1. This realtionship is expressed by the curve P=f (M) from the origin at 45° .

In Panel B of the figure, the inverse relation between the quantity of money and the value of money is depicted where the value of money is taken on the vertical axis. When the quantity of money is M2, the value of money is 1/P2. But with the doubling of the quantity of money to M, the value of money becomes one-half of what it was before, 1/P. And with the quantity of money increasing by four-fold to M1, the value of money is reduced by 1/P1. This inverse relationship between the quantity of money and the value of money is shown by downward sloping curve 1/P=f (M).

#### Assumptions of the Theory

Fisher’s theory is based on the following assumptions:

1. P is a passive factor in the equation of exchange which is affected by the other factors.

2. The proportion of M’ to M remains constant.

3. V and V’ are assumed to be constant and are independent of changes in M and M’.

4. T also remains constant and is independent of other factors such as M, M’, V and V’.

5. It is assumed that the demand for money is proportional to the value of transactions.

6. The supply of money is assumed as an exogenously determined constant.

7. The theory is applicable in the long run.

8. It is based on the assumption of the existence of full employment in the economy.

#### Criticisms of the Theory

The Fisherian quantity theory has been subjected to severe criticisms by economists.

#### 1. Truism

According to Keynes, “The quantity theory of money is a truism.” Fisher’s equation of exchange is a simple truism because it states that the total quantity of money (MV+M’V’) paid for goods and services must equal their value (PT). But it cannot be accepted today that a certain percentage change in the quantity of money leads to the same percentage change in the price level.

#### 2. Other Things not Equal

The direct and proportionate relation between quantity of money and price level in Fisher’s equation is based on the assumption that “other things remain unchanged”. But in real life, V, V’ and T are not constant. Moreover, they are not independent of M, M’and P. Rather, all elements in Fisher’s equation are interrelated and interdependent. For instance, a change in M may cause a change in V. Consequently, the price level may change more in proportion to a change in the quantity of money. Similarly, a change in P may cause a change in M. Rise in the price level may necessitate the issue of more money. Moreover, the volume of transactions T is also affected by changes in P.

When prices rise or fall, the volume of business transactions also rises or falls. Further, the assumptions that the proportion M’ to M is constant, has not been borne out by facts. Not only this, M and M’ are not independent of T. An increase in the volume of business transactions requires an increase in the supply of money (M and M’).

#### 3. Constants Relate to Different Time

Prof. Halm criticises Fisher for multiplying M and V because M relates to a point ot time and V to a period of time . The former is a static concept and the latter a dynamic. It is, therefore, technically inconsistent to multiply two non-comparable factors.

#### 4. Fails to Measure Value of Money

Fisher’s equation does not measure the purchasing power of money but only cash transactions, that is, the volume of business transactions of all kinds or what Fisher calls the volume of trade in the community during a year. But the purchasing power of money (or value of money) relates to transactions for the purchase of goods and services for consumption. Thus the quantity theory fails to measure the value of money.

#### 5. Weak Theory

According to Crowther, the quantity theory is weak in many respects. First, it cannot explain ‘why’ there are fluctutations in the price level in the short run. Second, it gives undue importance to the price level as if changes in prices were the most critical and important phenomenon of the economic system. Third, it places a misleading emphasis on the quantity of money as the principal cause of changes in the price level during the trade cycle. Prices may not rise despite increase in the quantity of money during depression; and they may not decline with reduction in the quantity of money during boom. Further, low prices during depression are not caused by shortage of quantity of money, and high prices during prosperity are not caused by abundance of quantity of money. Thus, “the quantity theory is at best an imperfect guide to the causes of the trade cycle in the short period,” according to Crowther.

#### 6. Neglects Interest Rate

One of the main weaknesses of Fisher’s quantity theory of money is that it neglects the role of the rate of interest as one of the causative factors between money and prices. Fisher’s equation of exchange is related to an equilibrium situation in which rate of interest is independent of the quantity of money.

#### 7. Unrealistic Assumption

Keynes in his General Theory severely criticised the Fisherian quantity theory of money for its unrealistic assumptions. First, the quantity theory of money is unrealistic because it analyses the relation between M and P in the long run. Thus it neglects the short run factors which influence this relationship. Second, Fisher’s equation holds good under the assumption of full employment. But Keynes regards full employment as a special situation. The general situation is one of the underemployment equilibrium. Third, Keynes does not believe that the relationship between the quantity of money and the price level is direct and proportional. Rather, it is an indirect one via the rate of interest and the level of output. According to Keynes, “So long as there is unemployment, output and employment will change in the same proportion as the quantity of money, and when there is full employment, prices will change in the same proportion as the quantity of money.” Thus Keynes integrated the theory of output with value theory and monetary theory and criticised Fisher for dividing economics “into two compartments with no doors and windows between the theory of value and theory of money and prices.”

#### 8. V not Constant

Further, Keynes pointed out that when there is underemployment equilibrium, the velocity of circulation of money V is highly unstable and would change with changes in the stock of money or money income. Thus it was unrealistic for Fisher to assume V to be constant and independent of M.

#### 9. Neglects Store of Value Function

Another weakness of the quantity theory of money is that it concentrates on the supply of money and assumes the demand for money to be constant. In order words, it neglects the store-of-value function of money and considers only the medium-of-exchange function of money. Thus the theory is one-sided.

#### 10. Neglects Real Balance Effect

Don Patinkin has critcised Fisher for failure to make use of the real balance effect, that is, the real value of cash balances. A fall in the price level raises the real value of cash balances which leads to increased spending and hence to rise in income, output and employment in the economy. According to Patinkin, Fisher gives undue importance to the quantity of money and neglects the role of real money balances.

#### 11. Static

Fisher’s theory is static in nature because of its such unrealistic assumptions as long run, full employment, etc. It is, therefore, not applicable to a modern dynamic economy.

### THE CAMBRIDGE EQUATIONS : THE CASH BALANCES APPROACH

As an alternative to Fisher’s quantity theory of money, Cambridge economists Marshall, Pigou, Robertson and Keynes formulated the cash balances approach. Like value theory, they regarded the determination of value of money in terms of supply and demand. Robertson wrote in this connection: “Money is only one of the many economic things. Its value, therefore, is primarily determined by exactly the same two factors as determine the value of any other thing, namely, the conditions of demand for it, and the quantity of it available.”

The supply of money is exogenously determined at a point of time by the banking system. Therefore, the concept of velocity of circulation is altogether discarded in the cash balances approach because it ‘obscures the motives and decisions of people behind it’. On the other hand, the concept of demand for money plays the major role in determining the value of money.

The demand for money is the demand to hold cash balances for transactions and precautionary motives.Thus the cash balances approach considers the demand for money not as a medium of exchange but as a store of value. Robertson expressed this distinction as money “on the wings” and money “sitting”. It is “money sitting” that reflects the demand for money in the Cambridge equations. The Cambridge equations show that given the supply of money at a point of time, the value of money is determined by the demand for cash balances. When the demand for money increases, people will reduce their expenditures on goods and services in order to have larger cash holdings.Reduced demand for goods and services will bring down the price level and raise the value of money. On the contrary, fall in the demand for money will raise the price level and lower the value of money. The Cambridge cash balances equations of Marshall, Pigou, Robertson and Keynes are discussed as under:

Marshall’s Equation. Marshall did not put his theory in equation form and it was for his followers to explain it algebraically. Friedman has explained Marshall’s views thus: “As a first approximation, we may suppose that the amount one wants to hold bears some relation to one’s income, since that determines the volume of purchases and sales in which one is engaged. We then add up the cash balances held by all holders of money in the community and express the total as a fraction of their total income.” Thus we can write:

M = kPY

where M stands for the exogenously determined supply of money, k is the fraction of the real money income (PY) which people wish to hold in cash and demand deposits, P is the price level, and Y is the aggreagate real income of the community. Thus the price level P= M/kY or the value of money (the reciprocal of price level) is 1/P = kY/M.

#### Pigou’s Equation

Pigou was the first Cambridge economist to express the cash balances approach in the form of an equation

P= kR/M.

where P is the purchasing power of money or the value of money (the reciprocal of the price level), k is the proportion of total real resources or income (R) which people wish to hold in the form of titles to legal tender, R is the total resources (expressed in terms of wheat), or real income, and M refers to the number of actual units of legal tender money. The demand for money, according to Pigou,consists not only of legal money or cash but also bank notes and bank balances. In order to include bank notes and bank balances in the demand for money, Pigou modifies his equation as

where c is the proportion of total real income actually held by people in legal tender including token coins, (1-c) is the proportion kept in bank notes and bank balances, and h is the proportion of actual legal tender that bankers keep against the notes and balances held by their customers.

Pigou points out that when k and R in the equation P=kR/M and k, R, c and h are taken as constants then the two equations give the demand curve for legal tender as a rectangular hyperbola. This implies that the demand curve for money has a uniform unitary elasticity. This is shown in Figure 2 where DD1 is the demand curve for money and Q1M1, Q2 M2, and Q3M3 are the supply curves of money drawn on the assumption that the supply of money is fixed at a point of time. The value of money or Pigou’s purchasing power of money P is taken on the vertical axis. The figure shows that when the supply of money increases from OM1 to OM2, the value of money is reduced from OP1 to OP2. The fall in the value of money by P1 P2 exactly equals the increase in the supply of money by M1M2. If the supply of money increases three times from OM1 to OM3, the value ofmoney is reduced by exactly one-third from OP1 to OP3. Thus the demand curve for money DD1 is a rectangular hyperbola because it shows changes in the value of money exactly in reverse proportion to the supply of money.

#### Robertson’s Equation

To determine the value of money or its reciprocal the price level, Robertson formulated an equation similar to that of Pigou.

The only difference between the two being that instead of Pigou’s total real resources R, Robertson gave the volume of total transactions T. The Robertsonian equation is M = PkT or where P is the price level, M is the total quantity of money, k is the proportion of the total amount of goods and services (T) which people wish to hold in the form of cash balances, and T is the total volume of goods and services purchased during a year by the community.

If we take P as the value of money instead of the price level as in Pigou’s equation, then Robertson’s equation exactly resembles Pigou’s P = kT/M.

#### Keynes’s Equation

Keynes in his A Tract on Monetary Reform (1923) gave his Real Balances Quantity Equation as an improvement over the other Cambridge equations. According to him, people always want to have some purchasing power to finance their day to day transactions. The amount of purchasing power (or demand for money) depends partly on their tastes and habits, and partly on their wealth. Given the tastes, habits, and wealth of the people, their desire to hold money is given. This demand for money is measured by consumption units. A consumption unit is expressed as a basket of standard articles of consumption or other objects of expenditure.

If k is the number of consumption units in the form of cash, n is the total currency in circulation, and p is the price for consumption unit, then the equation is n = pk. If k is constant, a proportionate increase in n (quantity of money) will lead to a proportionate increase in p (price level). This equation can be expanded by taking into account bank deposits. Let k’ be the number of consumption units in the form of bank deposits, and r the cash reserve ratio of banks, then the expanded equation is n = p (k + rk’)

Again, if k, k’ and r are constant, p will change in exact proportion to the change in n.

Keynes regards his equation superior to other cash balances equations. The other equations fail to point how the price level (p) can be regulated.

Since the cash balances (k) held by the people are outside the control of the monetary authority, p can be regulated by controlling n and r. It is also possible to regulate bank deposits k’ by appropriate changes in the bank rate. So p can be controlled by making appropriate changes in n, r and k’ so as to offset changes in k.

#### Criticisms of the Cash Balance Approach

The cash balances approach to the quantity theory of money has been criticised on the following counts:

#### 1. Truisms

Like the transactions equation, the cash balances equations are truisms. Take any Cambridge equation: Marshall’s P=M/kY or Pigou’s P=kR/M or Robertson’s P=M/kT or Keynes’s p=n/k, it establishes a proportionate relation between quantity of money and price level.

#### 2. Price Level does not Measure Purchasing Power.

Keynes in his A Treatise on Money (1930) criticised Pigou’s cash balances equation and also his own real balances equation. He pointed out that measuring the price level in wheat, as Pigou did or in terms of consumption units, as Keynes himself did, was a serious defect. The price level in both equations does not measure the purchasing power of money. Measuring the price level in consumption units implies that cash deposits are used only for expenditure on current consumption. But in fact they are held for”a vast multiplicity of business and personal purposes.” By ignoring these aspects, the Cambridge economists have committed a serious mistake.

#### 3. More Importance to Total Deposits

Another defect of the Cambridge equation “lies in its applying to the total deposits considerations which are primarily relevant only to the income deposits.” And the importance attached to k “is misleading when it is extended beyond the income deposits.”

#### 4. Neglects other Factors

Further, the cash balances equation does not tell about changes in the price level due to changes in the proportions in which deposits are held for income, business and savings purposes.

#### 5. Neglect of Saving-Investment Effect

Moreover, it fails to analyse variations in the price level due to saving-investment inequality in the economy.

#### 6. k and Y not Constant

The Cambridge equation, like the transactions equation, assumes k and Y (or R or T) as constant. This is unrealistic because it is not essential that the cash balances (k) and the income of the people (Y) should remain constant even during the short period.

#### 7. Fails to Explain Dynamic Behaviour of Prices

The theory argues that changes in the total quantity of money influence the general price level equiproportionally. But the fact is that the quantity of money influences the price level in an essential erratic and unpredictable way. Further, it fails to point out the extent of change in the price level as a result of a given change in the quantity of money in the short period. Thus it fails to explain the dynamic behaviour of prices.

#### 8. Neglects Interest Rate

The cash balances approach is also weak in that it ignores other influences, such as the rate of interest which exerts a decisive and significant influence upon the price level. As pointed out by Keynes, the relation between quantity of money and price level is not direct but indirect via the rate of interest, investment, output, employment and income. This is what the Cambridge equation ignores and hence fails to integrate monetary theory with the theory of value and output.

#### 9. Demand for Money not Interest Inelastic

The neglect of the rate of interest as a causative factor between the quantity of money and the price level led to the assumption that the demand for money is interest inelastic. It means that money performs only the function of medium of exchange and does not possess any utility of its own, such as store of value.

#### 10. Neglect of Goods Market

Further, the omission of the influence of the rate of interest in the cash balances approach led to the failure of neoclassical economists to recognise the interdependence between the commodity and money markets. According to Patinkin, “They laid an undue concentration on the money market a corresponding neglect of the commodity markets, and a resulting ‘dehumanising’ of the analysis of the effects of monetary changes.”

#### 11. Neglects Real Balance Effect

Patinkin has criticised the Cambridge economists for their failure to integrate the goods market and the money market. This is borne out by the dichotomy which they maintain between the two markets. The dichotomisation implies that the absolute price level in the economy is determined by the demand and supply of money, and the relative price level is determined by the demand and supply of goods.

The cash balances approach keeps the two markets rigidly apart. For instance, this approach tells that an increase in the quantity of money leads to an increase in the absolute price level but exercises no influence on the market for goods. This is because of the failure of Cambridge economists to recognise “the real balance effect.” The real balance effect shows that a change in the absolute price level does influence the demand and supply of goods. The weakness of cash balances approach lies in ignoring this.

#### 12. Elasticity of Demand for Money not Unity.

The cash balances theory establishes that the elasticity of demand for money is unity which implies that the increase in the demand for money leads to a proportionate decrease in the price level. Patinkin holds that “the Cambridge function does not imply uniform elasticity.” According to him, this is because of the failure of Cambridge economists to recognise the full implications of the “real balance effect”. Patinkin argues that a change in the price level will cause a real balance effect. For instance, a fall in the price level will increase the real value of cash balances held by the people. So when there is excess demand for money, the demand for goods and services is reduced. In this case, the real balance effect will not cause a proportionate but non-proportionate change in the demand for money. Thus the elasticity of demand for money will not be unity.

#### 13. Neglects Speculative Demand for Money

Another serious weakness of cash balances approach is its failure to consider the speculative demand for money. The neglect of the speculative demand for cash balances makes the demand for money exclusively dependent on money income thereby again neglecting the role of the rate of interest and the store of value function of money.