In
economics, the
equation of
exchange is the relation:
- M\cdot V = P\cdot 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:
- M\cdot V_T = P\cdot 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
- M\cdot V_T =\sum_{i} (p_i\cdot
q_i)=\mathbf{p}^\mathrm{T}\cdot\mathbf{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
- M\cdot V_T = P\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 \mathbf{p} may be
extracted as the multiplier P\,:
- M\cdot V_T = P\cdot
(\mathbf{p}_{real}^\mathrm{T}\cdot\mathbf{q}) = P\cdot T
and likewise for
- M\cdot V = P\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=\frac{M\cdot 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
Economists
Alfred Marshall,
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 k, a portion of
nominal income (nY).
(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:
- M_{D}=k\cdot nY
which, given the classical dichotomy and that
real income must equal
expenditures Q, is equivalent to
- M_{D}=k\cdot P\cdot Q
Assuming that the economy is at equilibrium (M_{D} = M), 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:
- M\cdot\frac{1}{k} = P\cdot Q
The money demand function is often conceptualized in terms of a
liquidity function, L(r,Y),
- M_D=P\cdot L(r,Y)
where Y is real income and r is the real
rate of interest. If V is taken to be a
function of r, then in equilibrium
- L(r,Q)=\frac{Q}{V(r)}
History
The equation of exchange was stated by
John Stuart Mill who expanded on the ideas
of
David Hume.
See also
Irving Fisher#Economic
theories
Notes
- Froyen, Richard T. Macroeconomics: Theories and
Policies. 3rd Edition. Macmillan Publishing Company: New York,
1990. p. 70-71.
- Mill, John Stuart; Principles of Political Economy
(1848).
- Hume, David; “Of Interest” in Essays Moral and
Political.
References