Equation_of_exchange

In economics, the equation of exchange is the relation:

$M\backslash cdot\; V\; =\; P\backslash cdot\; Q$

where, for a given period,

$M\backslash ,$ is the total amount of money in circulation on average in an economy.

$V\backslash ,$ is the velocity of money, that is the average frequency with which a unit of money is spent.

$P\backslash ,$ is the price level.

$Q\backslash ,$ is an index of expenditures.

In practice, $V\backslash ,$ 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:

$M\backslash cdot\; V\_T\; =\; P\backslash cdot\; T$

where

$V\_T\backslash ,$ is the transactions\' velocity of money, that is the average frequency across all transactions with which a unit of money is spent.

$T\backslash ,$ is an index of the real value of aggregate transactions.

Contents 1 Foundation 2 Applications 2.1 Quantity theory of money 2.2 Money demand 3 History 4 Notes 5 References

Foundation

The foundation of the equation of exchange is the more complex relation

$M\backslash cdot\; V\_T\; =\backslash sum\_\{i\}\; (p\_i\backslash cdot\; q\_i)=\backslash mathbf\{p\}^\backslash mathrm\{T\}\backslash cdot\backslash mathbf\{q\}$

where

$p\_i\backslash ,$ and $q\_i\backslash ,$ are the respective price and quantity of the i-th transaction.

$\backslash mathbf\{p\}$ is a vector of the $p\_i\backslash ,$.

$\backslash mathbf\{q\}$ is a vector of the $q\_i\backslash ,$.

The equation

$M\backslash cdot\; V\_T\; =\; P\backslash cdot\; T$

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 $\backslash mathbf\{p\}$ may be extracted as the multiplier $P\backslash ,$:

$M\backslash cdot\; V\_T\; =\; P\backslash cdot\; (\backslash mathbf\{p\}\_\{real\}^\backslash mathrm\{T\}\backslash cdot\backslash mathbf\{q\})\; =\; P\backslash cdot\; T$

and likewise for

$M\backslash cdot\; V\; =\; P\backslash cdot\; Q$

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

$P=\backslash frac\{M\backslash cdot\; V\}\{Q\}$

If $V$ and $Q$ were constant, then:

$\backslash frac\{d\; P\}\{P\}=\; \backslash frac\{d\; M\}\{M\}$

and thus

$\backslash frac\{d\; P/P\}\{d\; t\}=\backslash frac\{d\; M/M\}\{d\; t\}$

where

$t\backslash ,$ 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 $\backslash frac\{d\; M/M\}\{d\; t\}$.

Money demand

The equation can also serve as a basis for a money demand function:

$M\_D=\backslash frac\{P\backslash cdot\; Q\}\; \{V(R)\}=P\backslash cdot\; L(R,Q)$

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 MillMill, 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," The New Palgrave: A Dictionary of Economics, v. 2, pp. 175-77.
• Milton Friedman (1987. “quantity theory of money”, in The New Palgrave: A Dictionary of Economics ), v. 4, pp. 3-20.

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