Equation of exchange

In economics, the equation of exchange is the relation::$Mcdot\; V\; =\; Pcdot\; Q$where, for a given period,:$M,$ is the total amount of money in circulation on average in an economy.:$V,$ is the velocity of money, that is the average frequency with which a unit of money is spent.:$P,$ is the price level. :$Q,$ is an index of expenditures.In practice, $V,$ is calculated from values of the other terms.

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::$Mcdot\; V\_T\; =\; Pcdot\; T$where:$V\_T,$ is the transactions' velocity of money, that is the average frequency across all transactions with which a unit of money is spent.:$T,$ is an index of the real value of aggregate transactions.

Foundation

The foundation of the equation of exchange is the more complex relation:$Mcdot\; V\_T\; =sum\_\{i\}\; (p\_icdot\; q\_i)=mathbf\{p\}^mathrm\{T\}cdotmathbf\{q\}$where:$p\_i,$ and $q\_i,$ are the respective price and quantity of the "i"-th transaction.:$mathbf\{p\}$ is a vector of the $p\_i,$.:$mathbf\{q\}$ is a vector of the $q\_i,$.The equation :$Mcdot\; V\_T\; =\; Pcdot\; T$ 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 $mathbf\{p\}$ may be extracted as the multiplier $P,$: :$Mcdot\; V\_T\; =\; Pcdot\; (mathbf\{p\}\_\{real\}^mathrm\{T\}cdotmathbf\{q\})\; =\; Pcdot\; T$and likewise for:$Mcdot\; V\; =\; Pcdot\; Q$

Applications

Quantity theory of money

The quantity theory of money 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:$P=frac\{Mcdot\; V\}\{Q\}$If $V$ and $Q$ were constant, then::$frac\{d\; P\}\{P\}=\; frac\{d\; M\}\{M\}$ and thus:$frac\{d\; P/P\}\{d\; t\}=frac\{d\; M/M\}\{d\; t\}$where:$t,$ is time.That is to say that, if $V$ and $Q$ were constant, 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 $Q$ or of $V$ to $frac\{d\; M/M\}\{d\; t\}$.

Money demand

The equation can also serve as a basis for a money demand function::$M\_D=frac\{Pcdot\; Q\}\; \{V(R)\}=Pcdot\; L(R,Q)$

where the function $L(R,Q)$ is sometimes called the “liquidity function” or the demand for “real balances”, $M/P$.

History

The equation of exchange was stated by John Stuart Mill [Mill, John Stuart; "Principles of Political Economy" (1848).] who expanded on the ideas of David Hume. [Hume, David; “Of Interest” in "Essays Moral and Political".]

Notes

References

* Michael D. Bordo (1987). "equation of exchange," "", v. 2, pp. 175-77.
* Milton Friedman (1987. “quantity theory of money”, in "The " ), v. 4, pp. 3-20.