Equation of exchange
Encyclopedia
In economics
, the equation of exchange is the relation:
where, for a given period,
is the total nominal amount of money
in circulation on average in an economy.
is the velocity of money
, that is the average frequency with which a unit of money is spent.
is the price level
.
is an index of real expenditures (on newly produced goods and services).
Thus PQ is the level of nominal expenditures. This equation is a rearrangement of the definition of velocity: V = PQ / M. As such, without the introduction of any assumptions, it is a tautology. The quantity theory of money
adds assumptions about the money supply, the price level, and the effect of interest rates on velocity to create a theory about the causes of inflation and the effects of monetary policy.
In earlier analysis before the wide availability of the national income and product accounts
, the equation of exchange was more frequently expressed in transactions form:
where
is the transactions velocity of money
, that is the average frequency across all transactions with which a unit of money is spent (including not just expenditures on newly produced goods and services, but also purchases of used goods, financial transactions involving money, etc.).
is an index of the real value of aggregate transactions.
where
and 
are the respective price and quantity of the i-th transaction.
is a row vector of the 
.
is a column vector of the 
.
The equation
is based upon the presumption of the classical dichotomy
— that there is a relatively clean distinction between overall increases or decreases in prices and underlying, “real” economic variables — and that this distinction may be captured in terms of price indices
, so that inflationary
or deflationary components of p may be extracted as the multiplier P, which is the aggregate price level:
where
is a row vector of relative price
s; and likewise for
is most often expressed and explained in mainstream economics
by reference to the equation of exchange. For example a rudimentary theory could begin with the rearrangement
If
and 
were constant or growing at the same fixed rate as each other, then:
and thus
where
is time.
That is to say that, if
and 
were constant or growing at equal fixed rates, then the inflation rate would exactly equal the growth rate of the money supply.
An opponent of the quantity theory would not be bound to reject the equation of exchange, but could instead postulate offsetting responses (direct or indirect) of
or of 
to 
.
, A.C. Pigou, and John Maynard Keynes
, associated with Cambridge University, focusing on money demand instead of money supply, argued that a certain portion of the money supply will not be used for transactions, but instead it will be held for the convenience and security of having cash on hand. This proportion of cash is commonly represented as
, a portion of nominal income (
). (The Cambridge economists also thought wealth would play a role, but wealth is often omitted for simplicity.) The Cambridge equation
for demand for cash balances is thus:
which, given the classical dichotomy and that real income must equal expenditures
, is equivalent to
Assuming that the economy is at equilibrium (
), that real income is exogenous, and that k is fixed in the short run, the Cambridge equation is equivalent to the equation of exchange with velocity equal to the inverse of k:
The money demand function is often conceptualized in terms of a liquidity function,
,
where
is real income and 
is the real rate of interest
. If
is taken to be a function of 
, then in equilibrium
who expanded on the ideas of David Hume
. The algebraic formulation comes from Irving Fisher, 1911.
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
, the equation of exchange is the relation:
where, for a given period,
is the total nominal amount of moneyMoney supply
In economics, the money supply or money stock, is the total amount of money available in an economy at a specific time. There are several ways to define "money," but standard measures usually include currency in circulation and demand deposits .Money supply data are recorded and published, usually...
in circulation on average in an economy.
is the velocity of moneyVelocity of money
300px|thumb|Similar chart showing the velocity of a broader measure of money that covers M2 plus large institutional deposits, M3. The US no longer publishes official M3 measures, so the chart only runs through 2005....
, that is the average frequency with which a unit of money is spent.
is the price levelPrice level
A price level is a hypothetical measure of overall prices for some set of goods and services, in a given region during a given interval, normalized relative to some base set...
.
is an index of real expenditures (on newly produced goods and services).Thus PQ is the level of nominal expenditures. This equation is a rearrangement of the definition of velocity: V = PQ / M. As such, without the introduction of any assumptions, it is a tautology. The quantity theory of money
Quantity theory of money
In monetary economics, the quantity theory of money is the theory that money supply has a direct, proportional relationship with the price level....
adds assumptions about the money supply, the price level, and the effect of interest rates on velocity to create a theory about the causes of inflation and the effects of monetary policy.
In earlier analysis before the wide availability of the national income and product accounts
National Income and Product Accounts
The National Income and Product Accounts are part of the national accounts of the United States. They are produced by the Bureau of Economic Analysis of the Department of Commerce...
, the equation of exchange was more frequently expressed in transactions form:
where
is the transactions velocity of moneyVelocity of money
300px|thumb|Similar chart showing the velocity of a broader measure of money that covers M2 plus large institutional deposits, M3. The US no longer publishes official M3 measures, so the chart only runs through 2005....
, that is the average frequency across all transactions with which a unit of money is spent (including not just expenditures on newly produced goods and services, but also purchases of used goods, financial transactions involving money, etc.).
is an index of the real value of aggregate transactions.Foundation
The foundation of the equation of exchange is the more complex relationwhere
and are the respective price and quantity of the i-th transaction. is a row vector of the . is a column vector of the .The equation
is based upon the presumption of the classical dichotomy
Classical dichotomy
In macroeconomics, the classical dichotomy refers to an idea attributed to classical and pre-Keynesian economics that real and nominal variables can be analyzed separately...
— that there is a relatively clean distinction between overall increases or decreases in prices and underlying, “real” economic variables — and that this distinction may be captured in terms of price indices
Price index
A price index is a normalized average of prices for a given class of goods or services in a given region, during a given interval of time...
, so that inflationary
Inflation
In economics, inflation is a rise in the general level of prices of goods and services in an economy over a period of time.When the general price level rises, each unit of currency buys fewer goods and services. Consequently, inflation also reflects an erosion in the purchasing power of money – a...
or deflationary components of p may be extracted as the multiplier P, which is the aggregate price level:
where
is a row vector of relative priceRelative price
A relative price is the price of a commodity such as a good or service in terms of another; i.e., the ratio of two prices. A relative price may be expressed in terms of a ratio between any two prices or the ratio between the price of one particular good and a weighted average of all other goods...
s; and likewise for
Quantity theory of money
The quantity theory of moneyQuantity theory of money
In monetary economics, the quantity theory of money is the theory that money supply has a direct, proportional relationship with the price level....
is most often expressed and explained in mainstream economics
Mainstream economics
Mainstream economics is a loose term used to refer to widely-accepted economics as taught in prominent universities and in contrast to heterodox economics...
by reference to the equation of exchange. For example a rudimentary theory could begin with the rearrangement
If
and were constant or growing at the same fixed rate as each other, then:and thus
where
is time.That is to say that, if
and were constant or growing at equal fixed rates, then the inflation rate would exactly equal the growth rate of the money supply.An opponent of the quantity theory would not be bound to reject the equation of exchange, but could instead postulate offsetting responses (direct or indirect) of
or of to .Money demand
Economists Alfred MarshallAlfred Marshall
Alfred Marshall was an Englishman and one of the most influential economists of his time. His book, Principles of Economics , was the dominant economic textbook in England for many years...
, A.C. Pigou, and John Maynard Keynes
John Maynard Keynes
John Maynard Keynes, Baron Keynes of Tilton, CB FBA , was a British economist whose ideas have profoundly affected the theory and practice of modern macroeconomics, as well as the economic policies of governments...
, associated with Cambridge University, focusing on money demand instead of money supply, argued that a certain portion of the money supply will not be used for transactions, but instead it will be held for the convenience and security of having cash on hand. This proportion of cash is commonly represented as
, a portion of nominal income (). (The Cambridge economists also thought wealth would play a role, but wealth is often omitted for simplicity.) The Cambridge equationCambridge equation
The Cambridge equation formally represents the Cambridge cash-balance theory, an alternative approach to the classical quantity theory of money. Both quantity theories, Cambridge and classical, attempt to express a relationship among the amount of goods produced, the price level, amounts of money,...
for demand for cash balances is thus:
which, given the classical dichotomy and that real income must equal expenditures
, is equivalent toAssuming that the economy is at equilibrium (
), that real income is exogenous, and that k is fixed in the short run, the Cambridge equation is equivalent to the equation of exchange with velocity equal to the inverse of k:The money demand function is often conceptualized in terms of a liquidity function,
,where
is real income and is the real rate of interestInterest rate
An interest rate is the rate at which interest is paid by a borrower for the use of money that they borrow from a lender. For example, a small company borrows capital from a bank to buy new assets for their business, and in return the lender receives interest at a predetermined interest rate for...
. If
is taken to be a function of , then in equilibriumHistory
The equation of exchange was stated by John Stuart MillJohn Stuart Mill
John Stuart Mill was a British philosopher, economist and civil servant. An influential contributor to social theory, political theory, and political economy, his conception of liberty justified the freedom of the individual in opposition to unlimited state control. He was a proponent of...
who expanded on the ideas of David Hume
David Hume
David Hume was a Scottish philosopher, historian, economist, and essayist, known especially for his philosophical empiricism and skepticism. He was one of the most important figures in the history of Western philosophy and the Scottish Enlightenment...
. The algebraic formulation comes from Irving Fisher, 1911.