Question

# suppose that banks use x fraction of their deposits to buy treasuries. Derive the money multiplier...

suppose that banks use x fraction of their deposits to buy treasuries. Derive the money multiplier as a function of x, currency-deposit ratio cdr, and actual reserve ratio ar. How does an increase in x aﬀect money multiplier, monetary base, and money supply?

Let’s assume “M=money supply”, “C=currency hold by people”, “D=deposits”, “B=monetary base”, “R=reserve” and “T=treasuries”. The money supply and monetary base are given below.

=> “M = C + D”, and “B = C + R + T”.

=> M/B = (C + D) / (C+ R + T), => M/B = (c*D + D) / (c*D+ r*R + x*D), where “c=currency deposit ratio”, “r = reserve deposit ratio” and “x = fraction of deposit to buy treasuries”.

=> M/B = (c + 1) / (c + r + x), => M = m*B, where “m=money multiplier”.

So, as “x” increases implied more of deposit are used to buy treasuries, => “T” increases, => the monetary base also increases. Here “m” is the money multiplier where “x” on the denominator, => as “x” increases the money multiplier decreases. Now, as the money multiplier decreases the money supply also decreases, => “x” and money supply are inversely related to each other.

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