Wednesday, February 18, 2015

The Equation of Exchange, Its Versions, and Its Elements

Where does the value of money derive from? There is a common misconception that money must have some intrinsic value. A common question I hear from students is: “The dollar is backed by gold, right?” While the earliest moneys came into existence by virtue of the use value of the commodity traded, this is not the case in the world of modern finance.

As you know if you have been reading this series, money essentially arose by accident. When some agent comes in possession of a commodity that she plans only to exchange, rather than consume, the agent increases her demand for that good. The value gained from this type of increase in demand represents the value of the object as a medium of exchange. Thus, money has two sources of value: value derived from use and from exchange. As a commodity used as a medium of exchange gains value, producers are encouraged to increase production and new producers of the good are drawn by the opportunity of profit. Notice that endogeneity of the money stock (for example see this) – the tendency for the quantity of money to adjust to changes in demand for it – falls out of this example as a property that derives from the price system. More on this later. (This will also be important in coming posts when I review the automatic operation of a prototypical gold standard.) 

Like any good, money, at a given moment and over extended time horizons, is potentially offered for sale by agents at some array of prices. Ex post we observe the prices associated with particular transactions. Likewise, at a given moment, agents collectively demand some quantity of goods at a given array of prices. Ex post, these observed prices are equivalently the prices for supplied quantities (for simplicity let’s assume that the supplier of the good incurs all costs that might otherwise be spread over a network of producers).

Given the above construction, we can consider what might happen given changes in the preferences and agent knowledge concerning the available stock of money, its quantity demanded, and factors of supply. One additional tool for analysis is required: the equation of exchange (often identified as the quantity equation which is a derivative of the equation of exchange). The equation of exchange is an accounting identity that equates the quantity of money demanded with that supplied. The stock of money and its average velocity of circulation (an unfortunate term meaning the number of times the average currency unit is spent in a given time period) is identical to the quantity of goods purchased times their transaction prices. The rendition has thus far built upon microfoundations. In the modern formulation, we reduce the series of prices and goods to aggregate variables. Thus:

MV = Py


M = money stock
V = velocity
P = price level
y = real income

These variables tell us nothing of the composition of the economic system. Rather, it conveys an important truth about the supply of money (MV) and the demand for money (Py). As John Stuart Mill expressed in his Principles of Political Economy,

The supply of money, then, is the quantity of it which people are wanting to lay out; that is, all the money they have in their possession, except what they are hoarding, or at least keeping by them as reserve for future contingencies. The supply of money, in short, is all the money in circulation at the time.

The demand for money, again, consists of all the goods offered for sale. Every seller of goods is a buyer of money, and the goods he brings with him constitute his demand. The demand for money differs from the demand for other things in this, that it is limited only by the means of the purchaser.

As those demanding dollars and those supplying dollars successfully exchange and satisfy their preferences for dollars and goods, each market for money tends toward an equilibrium. This in no way suggests that the economy moves smoothly from equilibrium to equilibrium as agent actions and preferences are not independent of one another. The actions of every agent changes the distribution and prices of scarce goods. All markets, including markets for money, experience endogenous turbulence.

Types of Demand for Money 

It is necessary to more closely examine elements contained within the equation of exchange in terms of the demand for and supply of money. There are two types of demand for money contained within the identity. These are transactions demand and portfolio demand. MV represents the money stock that is available for exchange and that Py represents the demand for money by sellers of goods and services. Demand, however, can be broken down further. On the right hand side there is nominal income which is equivalently transactions demand. All goods sold must be exchanged for money. The money exchanged comprises transactions demand.

Portfolio demand for money is a different type of demand for money that consists of the demand to hold money. This demand is represented indirectly by the velocity, or rapidity, of circulation of money. Remember that the available supply of money is MV. The total money stock multiplied by the average number of times a given currency unit is spent represents the available money stock for a given time period. Money that is spent multiple times in a given period influences prices across that period. Money that is withheld from circulation by agents fails to positively impact the prices of any good, assuming that there is no expectation of expenditure later. The equation of exchange can be rewritten to include this demand for money on the right hand side of the identity.

MV = Py

M = (Py)/V

Let k = 1/V

M = Pyk

This formulation, known as the Cambridge cash-balance version, shows that the total money stock is equal to the product of transactions demand (Py) and demand to hold money on reserve (k).

The Quantity Theory

The quantity theory is a particular rendition of the equation of exchange where velocity – the inverse of portfolio demand – is assumed constant and where real income is thought not to be affected in the long run by changes in the money stock.  It is stated as

This is a long run definition that shows that an increase in the money stock, all else equal, will lead to an increase in prices generally.

Endogenous Money

With the quantity theory, causation is thought to work from the left to the right side of the equation, but this is not always the case. Imagine that there occurs a general increase in demand to hold money. The immediate effect may be for prices to fall, but this does not occur instantaneously. Remember that an increase in demand for money tends to increase its price, also known as purchasing power. Under a commodity money standard, a rise in the price of money will encourage production of that money. This is easily conceptualized with the Cambridge cash-balance interpretation. Recall

               M = Pyk

If k – portfolio demand for money – increases, then M may also increase, owing to an increase in money’s price, in order to offset this effect.

Endogenous Credit

An increase in demand for money does not have to be offset by an increase in the base. Thus far, I have not differentiated between different types of money. Money consists not only of base money, but of higher level moneys known is fiduciary moneys. (Fiduciary meaning the value depends upon faith in the promise to repay). These moneys are linked to base moneys typically by a promise of repayment. Into the first half of the 20th century, this type of currency was best embodied by deposit slips which served as a claim to base money and which were exchanged as money. Then, as now, the money deposited at the bank was lent to agents who demand money in the present and promise repayment in the future. Their demand for money now is constrained by the price of money in the future known as the interest rate. If agents generally increase their demand to hold money, other agents who need money for transactions in the immediate future can borrow and thereby increase the quantity of money presently available. Of course, they must be willing to repay this debt and the interest accrued during the life the loan. As we have not yet covered banking, it will suffice for you to remember that credit adjusts according to changes in demand for money. This will be discussed in greater detail in later posts.

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