The simple model

• M1 = CU + D
  • money supply (M1) = currency in hands of public (CU) + balances in deposit accounts (D)
• M _h = CU + R
  • high powered money [M _h (coins & currency)] = cash in hands of public (CU) and cash in banks (R)
• CU = 0
  • public holds no currency (CU)
• R = r_r*D
  • reserves (R) are proportional to deposit balances (D)

Combining these equations we get:

• M1/M _h = (CU + D) / (CU + R) = D / R
• M1/M _h = (R/r_r)/R = (1/r_r)

An extended money multiplier model

• M1 = CU + D
  • money supply (M1) = currency in hands of public (CU) + balances in deposit accounts (D)
• M _h = CU + R
  • high powered money [M _h (coins & currency)] = cash in hands of public (CU) and cash in banks (R)
• CU = c_d D
  • public's holding of currency (CU) is proportional to their holding of deposit balances (D)
• R_r = r_r*D
  • required reserves (R_r) are proportional to deposit balances (D)
• R_t = R_r + R_e
  • Total reserves (R_t) = required reserves (R_r) + excess reserves (R_e)

Combining these equations we get:

• M_h = CU + R = c_d D + r_d D = (c_d + r_d ) D
• M1 = CU + D = c_d D + D = ( c_d + 1 ) D
• M1/M_h = ( c_d + 1 ) D / (c_d + r_d ) D
  M1 = [( c_d + 1 ) / (c_d + r_r + r_e ) ] M_h

• decrease in reserve rate
  • decline in required reserve rate r_r
  • decline in excess reserve rate r_e
• decrease in desire for cash c_d
• increase in high-powered money M_h