Mathematical note

The mathematical note derives the formula for the money multiplier.

• Money is M.

M = Deposits + Currency

• The monetary base is MB.

MB = Currency+ Reserves.

· Currency = a ´ Deposits. The fraction “a” is the currency drain ratio, which is equal to C/D where C is currency and D is deposits..

· Reserves = b ´ Deposits. The fraction “b” is the reserve ratio, which is equal to R/D where R is reserves..

• Using these definitions for desired currency holdings and desired reserves shows that:

M = Deposits + a ´ Deposits = (1 + a) ´ Deposits

MB = a ´ Deposits + b ´ Deposits = (a + b) ´ Deposits

• Using “D” for “change in,” these last two equalities show that:

DM = (1 + a) ´ DDeposits

DMB = (a + b) ´ DDeposits

• The money multiplier is the ratio of DM to DMB. So dividing the top equation by the bottom gives:

Money multiplier = .

• The formula shows that the size of the money multiplier depends on the reserve ratio and the currency drain.
• In 2008, when times were more “normal,” for M1 the currency drain ratio, a, was equal to 1.24 and the reserve ratio, b, was equal to 0.28, so the money multiplier for M1 was = 1.47. In 2008, for M2 the currency drain ratio, a, was equal to 0.12 and the reserve ratio, b, was equal to 0.03, so the money multiplier for M2 was = 7.5.
• In 2010, when times banks are holding substantially more reserves than in past decades, for M1 the currency drain ratio, a, is equal to 1.06 and the reserve ratio, b, is equal to 1.32, so the money multiplier for M1 was = 0.87. In 2010, for M2 the currency drain ratio, a, is equal to 0.11 and the reserve ratio, b, is equal to 0.14, so the money multiplier for M2 was = 4.44.

Дата добавления: 2015-11-04