The Money Multiplier
The simple model says one dollar of reserves becomes 1/rr of deposits. Add currency that the public holds (c) and reserves banks keep beyond the requirement (e) and the leakage shrinks the true multiplier to m = (1 + c) / (rr + e + c).
Simple multiplier 1/rr10.00×
Full multiplier m2.80×
Gap (overstatement)7.20×
ΔMoney from $1,000 reserves$2,800
Deposit-expansion cascade · $1,000 of new reserves
Each round, 35.7% leaks out as required reserves, excess reserves and currency, so the loan shrinks. Money created after round 8: $82 still to come.
Sum of all rounds → m × $1,000$2,800
Try this. Set c = 0 and e = 0: the full multiplier collapses onto the simple 1/rr. Now push c toward 1 (a cash-heavy crisis) and watch m fall far below 1/rr even though rr never changed.