• The simple model
  • M1 = CU + D
  • M _h = CU + R
  • CU = 0
  • R = r_r*D
  • M1/M _h = (CU + D) / (CU + R)
  • M1/M _h = (R/r_r)/R = (1/r_r)

• r_r = required reserve rate
• CU = currency holdings
• D = deposit account balances
• R = reserves held in banks
• M1 = money supply
• Mh = high powered money

• An extended money multiplier model
  • allow banks to hold excess reserves
  • allow households/firms to hold cash
  • Model
  • r_d =R/D iff R = r_d D
  • r_d = r_r + r_e
  • c_d =CU/D iff CU = c_d D
  • 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

  Where

  • r_d = reserve rate
  • r_r = required reserve rate
  • r_r = excess reserve rate
  • c_d = currency holdings /demand deposit (cash preference)
  • M_h = high powered money
  • M1 = money supply

• Money supply increases with:
  • 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

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