The second question is the most important. Suppose M1 is given by currency plus
ID: 1202835 • Letter: T
Question
The second question is the most important.
Suppose M1 is given by currency plus deposits: M1 = C + D, and the base is given by currency plus reserves: B = C + R. Recall that the base and the money supply are connected by the multiplier, m, such that B * m = M1. Suppose further that banks must keep a fraction, rr, of their deposits on hand R = rr * D, and the public holds a fraction, k, of their deposits as cash so C = k * D. Solve for the multiplier in terms of just r and k. Refer to the above. Recessions historically have been (and sometimes still are) accompanied by runs on banks. What complications are presented for monetary policy in this situation? What happens if the central bank does nothing?Explanation / Answer
Answer (1)
Given M1 = C + D ...................(1)
B = C + R ...................(2)
The relationship between M and B can be expressed as the ratio of M to B. So divide the equation (1) by (2):
M1 / B = (C + D) / (C + R) ...........(3)
Divide the numerator and denomenator of the right hand side of the equation (3) by D:
M1 / B = [(C /D) + (D / D)] / [(C / D) + (R / D)]
M1 / B = [1 + (C / D)] / [(C / D) + (R / D)] ...........(4)
By substituting k for (C / D) and rr for R / D, equation (4) becomes
M1 / B = (1 + k) / ( k +rr) .....................(5)
Money Supply (M1) = (1 + k) / ( k +rr) * B .....................(6)
Money Supply (M1) = m * B .....................(7)
The quotient of equation (7) is the money multipluier m. Thus = (1 + k) / ( k +rr)...............(8)
Now the relatioship between the money supply (M1) and base (B) of equation (7) becomes B * m = M1.