Sunday, November 03, 2013

On Money Multiplier

Central bank prints yen bills, say 1 billion yen, by buying government bonds worth 1billion yen from private banks. Then private banks lend 1 billion yen to people.

Through this process, how much money can increase in the economy as a whole?

This is the problem of money multiplier and here's how to calculate it:

1.People borrow 1 billion yen from private banks and they deposit 1/(1+a) billion yen in the private banks. a is a ratio of currency to deposit, currency/deposit, or deposit:currency=1:a.

2.Then the private banks again lend (1-r)/(1+a) billion yen to the people and they again deposit (1-r)/(1+a)^2 billion yen in the private banks. r is a ratio of reserve to deposit, reserve/deposit. Reserve is the deposit of private banks in the central bank.

Through these processes repeated again and again, money in the economy as a whole increases by (1+a)/(r+a) billion yen.

The calculation is,  1+1*(1-r)/(1+a)+1*{(1-r)/(1+a)}^2+...= 1*(1+a)/(r+a)

The right term, (1+a)/(r+a), in the above identity is called money multiplier.

For example, if a=0.4,r=0.2, then 7/3=2.333, and thus 1billion yen can increase by 2.3 billion yen (1.67 billion yen increase in deposit and 0.63 billion yen increase in currency).

If a increases to 0.5, r remains the same, 0.2, then the money multiplier is 2.14. If a remains the same 0.4, r increases to 0.3, then the money multiplier is just 2.

The lesson is, if the ratio of currency to deposit or the ratio of reserve to deposit increases, the money multiplier decreases. 

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