Marshall’s k

Marshall’s k (Marshallian k) refers to the proportion of money holdings relative to an economy’s nominal income. In other words, it shows how much money people choose to hold compared to the total nominal income of the economy.

In an economy where nominal national income over a given period is PY, and the velocity of money is V, the amount of money needed for real transactions—that is, the demand for money—can be expressed as follows.

According to the quantity equation of exchange,

MV = PY,
which can be rearranged as
M = (1 / V) × PY = kPY.

From this relationship, Marshall’s k is defined as the inverse of the velocity of money. It is treated as a constant under the assumption that the velocity of money remains stable.

However, in reality, the velocity of money is influenced by various institutional factors, such as the development of payment systems and financial technologies. It can also fluctuate significantly in the short run.

During economic expansions, the velocity of money tends to increase, while during recessions it tends to slow down, showing a pro-cyclical pattern.

Marshall’s k is sometimes used as an indicator to assess whether the money supply of a country is appropriate relative to the size of its economy, based on the relationship

Ms = Md = kPY.

In practice, however, it is very difficult to determine an “optimal” level of Marshall’s k in advance. This is because money supply itself is defined in various ways (such as currency in circulation, M1, M2, or broader liquidity measures), and because financial innovation causes the velocity of money to change frequently, especially in the short term.