Equation of exchange

Overview

In monetary economics, the equation of exchange is the relation:

M ⋅ V = P ⋅ Q

{\displaystyle M\cdot V=P\cdot Q}

where, for a given period,

M

{\displaystyle M\,}

is the total money supply in circulation on average in an economy.

V

{\displaystyle V\,}

is the velocity of money, that is the average frequency with which a unit of money is spent.

P

{\displaystyle P\,}

is the price level.

Q

{\displaystyle Q\,}

is an index of real expenditures (on newly produced goods and services). The equation of exchange is a rearrangement of the definition of velocity. Velocity, or the average frequency with which money is spent, can be calculated as the ratio between the total amount of money spent,

P Q

{\displaystyle PQ}

, and the total amount of money that exists,

M

Foundation

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

M ⋅

V

T

=

∑

i

(

p

i

⋅

q

i

) =

p

T

⋅

q

{\displaystyle M\cdot V_{T}=\sum _{i}(p_{i}\cdot q_{i})=\mathbf {p} ^{\mathrm {T} }\cdot \mathbf {q} }

where:

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 =

M ⋅ V

Q

{\displaystyle P={\frac {M\cdot V}{Q}}}

If

V

{\displaystyle V}

and

Q

{\displaystyle Q}

were constant or growing at the same fixed rate as each other, then:

d P

P

=

d M

M

{\displaystyle {\frac {dP}{P}}={\frac {dM}{M}}}

and thus

d P

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

{\displaystyle k}

, a portion of nominal income (

n Y

{\displaystyle 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 ⋅ n Y

{\displaystyle M_{D}=k\cdot nY}

which, given the classical dichotomy and that real income must equal expenditures

Q

{\displaystyle Q}

, is equivalent to

M

D

= k ⋅ P ⋅ Q

{\displaystyle M_{D}=k\cdot P\cdot Q}

Assuming that the economy is at equilibrium (

Valuation of unconventional money

The equation of exchange has been used to derive the value of means of payment that are not used as units of account, such as cryptocurrencies. The derivation relies on two assumptions. Assumption 1. The number of cryptocurrency units required for any payment,

P

{\displaystyle P}

, depends only on the dollar amount that must be paid,

P

$

{\displaystyle P^{\$}}

, and the latest cryptocurrency price,

S

$

/

coin

{\displaystyle S^{\$/{\text{coin}}}}

, so that

P =

P

$

/

S

$

History

The equation of exchange was stated by John Stuart Mill who expanded on the ideas of David Hume. An early algebraic formulation comes from Simon Newcomb. The transaction version of the equation of exchange was popularized by Irving Fisher.

See also

Irving Fisher § Economic theories

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.