Fisher’s Equation of Exchange
The equation of exchange was the mechanism employed by Irving Fisher to analyze the relationship between money and economic activity.1 The basis for the equation was the proposition that there are two sides to every transaction: a buyer and a seller. Aggregating across all transactions, the total value of all things bought (B) is exactly equal to the total value of all things sold (S).
The right-hand side of the relationship is the goods side. The total value of all things sold (S) is equal to the sum of the price times the quantity for each item sold. Fisher wrote this as PT, where P is the average price and T is the total number of transactions. PT, then, is substituted for S on the right-hand side of equation 2.1.
The left-hand side is the money side. In a strictly monetary economy, all things are sold for money (M). However, it is not possible to substitute M for B on the left-hand side of equation 2.1 because it is very unlikely that each unit of money was used, on average, exactly one time in financing all exchanges (PT).
What relates money to spending is the velocity of circulation of money. Velocity (VT), or transactions velocity, is the average number of times each unit of money is used in financing PT. It is now possible to represent total expenditures for goods and services as MVT, and this term is substituted for B in equation 2.1. The result is Fisher’s version of the equation of exchange.
One problem with this version of the equation of exchange is that it is not operational. Fisher advanced the equation prior to the development of our system of national income accounts. Subsequent to the development of these accounts, Fisher’s equation was modified to a form that was operational.
where M is the money supply,
V is income velocity of money,
P is the average price of all final goods services sold, and
y is the number of final goods and services sold.
Modern governments measure both the money supply (M) and nominal gross domestic product (GDP). But, the right-hand side of 2.3 is nominal GDP. With measures for both M and Py, it is possible to calculate velocity (V). Consequently, this modified version of the equation of exchange is fully operational and, for that reason, generally is preferred to Fisher’s original statement of the equation of exchange.
Before proceeding, this modified version of the equation of exchange (which is subsequently employed) is compared to Fisher’s version. Py, or nominal GDP, is a measure of all final goods and services currently produced. Fisher’s PT is the total value of all goods and services currently exchanged. Because many goods and services currently exchanged are not currently produced, Py is a proper subset of PT, that is, every good or service currently produced is contained in PT. Many things currently exchanged (and in PT), however, were produced in the past. Hence, they are not included in GDP. One such example is the sale of a used automobile.
With the same money supply (M), and with Py different from PT, the velocity terms in equations 2.2 and 2.3 are not the same. For that reason, they are represented by different symbols. Fisher’s VT is the number of times (on average) money is used in all exchanges. V, on the other hand, is the average number of times each unit of money is used in the financing of GDP expenditures. Because Py is smaller than PT, V is smaller than VT.
Fisher’s equation of exchange was criticized on the grounds that it was a tautology. Fisher’s response was that just because a relationship is tautological does not mean it is without value. The equation of exchange is a case in point. Even though it is a tautology, it is a very useful device for analyzing factors responsible for changes in the purchasing power of money.
Based on the equation of exchange, all changes in the purchasing power of money are the result of changes in M, V, y, or in some combination of the three. The reason is that if P (and the purchasing power of money, 1/P) changes, something else in the equation of exchange must have changed as well. Otherwise, MV ≠ Py.
Increases in M bring about increases in P. Thus, they reduce the purchasing power of money (1/P). Increases in P can also result from a more frequent use of money. Hence, like M, V and the purchasing power of money are indirectly related. Increases in production (y), however, have the opposite effect. A greater production of goods and services, with the same amount of money, leads to lower prices for goods and services. That is the same thing as an increase in the exchange value of money.
To summarize, M and V are indirectly related to the purchasing power of money, while y is directly related. These relationships are shown in equation 2.4, which shows 1/P as derived from the equation of exchange.
Velocity and the Demand for Money
Cambridge University economists in England developed the theory of money in a manner that did not involve the equation of exchange. A.C. Pigou, for example, began his analysis by treating money like any other good. From his perspective, there is a market for money, and his approach was to analyze the supply and demand for money within that context. He was able to derive several money-spending relationships that were similar to those of Fisher.2
No detailed analysis of differences in the two approaches is undertaken here.3 However, it is important to note the relationship between the demand for money and Fisher’s velocity of circulation of money. The two are inversely related to one another. When individuals increase their demand for money balances, they tend to hold money for a longer period of time, and the velocity of circulation of money falls. Likewise, reductions in the demand for money tend to increase velocity. Individuals are spending money more frequently.
Derivation of the inverse relationship is shown in equations 2.5–2.7. Note that velocity is a relative measure of the demand for money. It reveals how much money individuals collectively demand in relationship to GDP. Moreover, the relationship is in inverse form.
In monetary equilibrium, the quantity of money supplied (M) is equal to the nominal quantity of money demanded (Md). Hence, Md can be substituted for M in equation 2.6.
A second adjustment in 2.6 is also appropriate. If rational economic agents think in real terms, they typically demand money balances expressed in terms of goods and services. Consequently, the numerator and denominator on the right-hand side of equation 2.6 are both divided by the average price (P). The resulting relationship between velocity and the demand for real money balances (Md/P) is stated in equation 2.7.
When individuals collectively demand more real money balances in relationship to real GDP (y), velocity falls. Alternatively, relative reductions in real money demand result in an increase in the velocity of circulation of money.
When V is viewed as a proxy for money demand, a somewhat different interpretation of Fisher’s equation of exchange (equation 2.3) is also implied. Money supply (M) and money demand (in inverse form V) interact to determine the level of nominal GDP. Either an increase in the money supply or a decrease in money demand causes money GDP to increase. A reduction in the money supply or a rise in money demand, bring about the opposite: a decrease in money GDP.
Money and the Economy
The equation of exchange is also useful in analyzing the relationship between money and the economy. If the money supply changes, the offsetting entry in the equation of exchange must be either V, P, or y (or some combination of the three). Those three possibilities are shown in Exhibit 2.1.
Exhibit 2.1 Absorption of Money Changes Within the Equation of Exchange
The Liquidity Trap
If increases in the money supply are absorbed in the form of a reduction in velocity, we are experiencing what John Maynard Keynes described as absolute liquidity preference. In this case, monetary policy has no effect on aggregate spending because individuals hold rather than spend any increase in the quantity of money. Keynes considered this case a theoretical curiosity: “Whilst this limiting case might become practically important in the future, I know of no example of it hitherto.”4
While Keynes may have discounted its importance, that was not the case for his disciples (Keynesians). Rejecting Keynes’ reticence, they renamed the phenomenon the liquidity trap and raised it to the level of a general case. The liquidity trap is something that routinely occurs during business cycle downturns. One of the more dramatic episodes, according to the Keynesians, was the Great Depression. During that cataclysmic decline, velocity decreases thwarted efforts by the Federal Reserve to end the depression through increases in the money supply.
When Keynesian economics attracted more followers in the 1950s and 1960s, the concept of the liquidity trap gained credibility. This had enormous policy implications. For, if monetary policy is unreliable, its policy role is necessarily a secondary one. The ensuing relegation of monetary policy to the background made possible the major acceleration of money growth and the Great Inflation that plagued the U.S. economy in the late 1960s and the 1970s.
The discovery that activist economic policies lead to more inflation, and not higher living standards, placed followers of Keynes on the defensive. As Keynesian economics lost credibility, so too, did the concept of the liquidity trap. Significant in its demise were research findings published by Milton Friedman and Anna Schwartz in A Monetary History of the United States.5
An important component of their study was the construction of a money supply series for the United States back to 1860. This data led Friedman and Schwartz to reject the Keynesian interpretation of the Great Depression. The U.S. economy was not caught in a liquidity trap in the 1930s. In contrast to Keynesian assertions, it was not a situation where individuals were holding rather than spending Fed-induced increases in the money supply. The data revealed, instead, an unprecedented decline in the stock of money, which fell by 35% from 1929–1933. With a lack of empirical support for the existence of a liquidity trap, the concept was unceremoniously relegated to the position originally assigned to it by Keynes.
Money and Real GDP
If changes in money are not absorbed by changes in velocity, they affect spending. Through their impact the right-hand side of the equation of exchange, they bring about changes in the level of nominal GDP. Such changes in nominal GDP can, in turn, be further decomposed into changes in the average price (P), real GDP (y), or both.
Economists such as Irving Fisher, R.G. Hawtrey, and more recently, Milton Friedman argue that money affects spending differently in the short-run than in the long-run. In the short-run, changes in money primarily affect real magnitudes such as the level of production (y) and the employment of resources. Moreover, changes in M are considered the principal cause of business cycle fluctuations. These economists have what is called a monetary theory of the trade cycle. With fiat money controlled by central banks, it is governments that are the primary source of economic instability. It follows that the best prospect for taming the business cycle is for governments to provide greater monetary stability.
The long-run impact of changes in money is primarily on prices. This money-price nexus is exhibited historically by numerous cases of government debasement of money. One of the earliest episodes involved the Roman emperor, Nero, but the tradition is a robust one. It transcends different types of money and different forms of government. This long-run relationship between money and prices is formally known as the quantity theory of money.
The Quantity Theory of Money
The quantity theory of money is a misnomer. It is not a theory of money. Rather, it is a theory about what determines the long-run purchasing power of money. Fisher employed his equation of exchange as a mechanism to explain the quantity theory. It deserves mention that the equation of exchange alone is a tautology and not a theory. Hence, it is not sufficient to state that M and P are directly related to one another in the equation of exchange, ceteris paribus. That too, is a tautology.
To develop his version of the quantity theory of money, Fisher augmented the equation of exchange with two additional assumptions. First, he argued that, in the long-run, changes in the money supply do not affect the velocity of circulation of money. Other factors, however, do. He cited urbanization, changes in commercial customs (such as the use of credit), and changes in technology (e.g., improved transport) as causing velocity to change in the long-run. Hence, velocity was not a constant. It just wasn’t affected by the money supply.
Second, in the long-run, changes in the money supply do not affect the level of production (y). If they did, we would have a guaranteed remedy for world poverty. Simply send printing presses to poor countries, and let them print massive quantities of fiat money. In fact, many of them have already tried this, but to no avail. The reason why this does not work is obvious. Printing additional fiat money does not increase wealth. By discouraging production, in many cases, it does just the opposite.
The observation that money does not affect production in the long-run does not imply that production is a constant. Long-run production obviously does change. According to Fisher, factors responsible for long-run increases in production are increased availability of resources and improvements in technology.6
Fisher’s two assumptions are summarized in equations 2.8 and 2.9. Changes in the quantity of money (dM) have no long-run effect on either the velocity of money or the quantity of real GDP. The change in velocity (dV) and the change in real GDP (dy) with respect to a change in money are both equal to zero.
If money does not affect either V or y, it must affect P. That gives rise to Fisher’s version of the quantity theory of money. Simply stated, in the long-run, a given change in the money supply occasions a direct and equi-proportionate change in the average price (relative to what it otherwise would have been).
Two things deserve mention here. First, while Fisher’s quantity theory of money is a tautology as a theoretical exercise, it is not so empirically. It is possible for long-run changes in money to be reflected in something other than the average price. They could potentially be absorbed by velocity changes or changes in production.
Second, Fisher’s theory does not state that long-run movements in M and P are equal to one another. The long-run movement in P is a resultant of the combined effects of changes in M, V, and y. While all three do have an influence, they are not all equally important. Fisher emphasized the relative importance of changes in money, which is a trademark of those in the quantity theory tradition.
A Dynamic Version of the Equation of Exchange
Hitherto, the analysis of money and the economy employed Fisher’s equation of exchange expressed in terms of levels. The dynamic version of the equation of exchange expresses that same relationship in terms of growth rates.
The following didactic exercises attempt to mimic historical episodes through the employment of this dynamic version of the equation of exchange.
Secular Deflation
The United States experienced secular deflation in the last one-third of the 19th century. This was a period of great innovations and rapid industrialization, and the U.S. growth rate for production was high by historical standards. The country was on the gold standard, and there were no new major discoveries of gold from mid-century until the late 1890s. As a consequence, production growth exceeded the growth rate for money. The result was a secular increase in the purchasing power of money.
Secular deflation of this type is portrayed as Case I in Exhibit 2.2. Real GDP is growing 4% per year, while the money supply is unchanged (dM/M = 0). Assuming no secular trend in velocity, the average price falls at the annual rate of 4%.
Exhibit 2.2 Dynamic Version of the Equation of Exchange
Case I has an important implication. It demonstrates that inflation is not a necessary condition for economic growth. To the contrary, by increasing the availability of goods and services, economic growth is actually deflationary. This deserves mention because the myth that inflation is necessary for economic growth was popular among development economists in the last half of the 20th century. Some contemporary economists and members of the popular media often expound a similar view. Alarmed when the U.S. inflation rate falls below 2%, they agonize about the potential dire consequences of deflation. The 19th century U.S. experience provides a good historical counterexample for those harboring such thoughts.
Secular Price Stability
For a country to experience secular price stability, money growth should approximate the long-run growth of production. That occurs in Case II, Exhibit 2.2. Both money growth and production growth are 4% per annum. Assuming no secular growth in velocity, the result is secular price stability.
Long-run price stability of this type does not imply that prices are stable every year. In some years, an economy might experience inflation; in others, deflation. For the entire period, however, the average price is stable. This was the U.S. experience with commodity and fiduciary monies. In 1933, the average price was approximately the same as it was in the early 1780s. (See Chapter 1, page 8, for a graph of the average price during this period.) Market forces, under these monetary standards, tend to bring about money growth rates that conform to the growth rate of production. That adjustment process was discussed on pp. 9–10.
The Quantity Theory of Money
In the dynamic version of the quantity theory of money, a given change in the growth rate of money (dM/M) occasions a direct and equi-proportionate change in the growth rate of the average price (dP/P) relative to what it otherwise would have been.
This dynamic version is illustrated by an economy that moves from Case II to Case III in Exhibit 2.2. The annual growth rate for money increases by 4%—from 4% percent in Case II to 8% in Case III. The ensuing rate of change in prices also increases by 4%, from 0% to 4% per year. While the quantity theory implies that the increase in the growth rate of the average price is equal the increase in the growth rate of money, note that it does not imply that the growth rates of money and the average price are the same. They are not in Case III.
As articulated by many in the late 19th century, inflation occurs when there is “too much money chasing too few goods.” Money is growing more rapidly than is production (Case III). In that sense, Case III reflects the secular inflation in the United States following the imposition of fiat money in 1933. It is the result of too much money chasing too few goods.
Hyperinflation
Hyperinflation occurs when prices increase extremely rapidly. There is no threshold inflation rate where hyperinflation commences. Nevertheless, hyperinflation is generally not difficult to identify. It occurs exclusively under fiat money regimes, and can last for more than one year. However, it is not a long-run phenomenon. The process typically comes to an end when the government responsible for the hyperinflation announces that it is undertaking monetary reform. The reform usually assumes the form of a new fiat money. It is physically different from the previous fiat money, and often has a new name, for example, the real as opposed to the peso. A rate of exchange of the old for the new currency is announced (x-“kazillion” units of the old for one unit of the new). There were numerous such episodes under government stewardship of money following the departure from fiduciary money. A recent case was Zimbabwe.
Case IV, Exhibit 2.2, is a numeric example of what happens during hyperinflations. The central bank increases the money supply at the rate of 1,500% a year. With the money side of the equation of exchange increasing at this rate, great stress is transferred to the right-hand side of that equation. That stress is dissipated in the form of massive increases in prices.
This stress on the goods side (or right-hand side) of the equation of exchange is exacerbated by the behavioral response of owners of money balances. With hyperinflation, transactions costs in the form of storage costs for money increase dramatically. Simultaneously, large changes in relative prices are occurring, and everyone becomes an inadvertent speculator. The kind of portfolio adjustments individuals make in this environment matters a great deal. One type of portfolio adjustment, however, is not speculative. The purchase of almost any type of good is preferred to the ownership of money, and virtually all individuals soon learn to minimize their holding period for money balances. The result is a massive increase in the velocity of circulation of money. In Case III, the increase is 500% per year.
Reflecting pressures from both M and V, the annual inflation rate is now 2,000%. Although not portrayed in this example, the disruptive influence of this degree of monetary instability often leads to a significant decline in production (y). While very small when compared to the influences of M and V, this fall in production further erodes the purchasing power of money.
With M, V, and y all exerting upward pressure on P, the real money supply (or M/P) actually falls during hyperinflations. For some who focus on the real money supply, this can lead to the perverse interpretation that the economy is experiencing “tight money.”