2. Take the example of Silas depositing his $1,000 in cash into First Street Bank and assume that the required reserve ratio is 10%. But now assume that each time someone receives a bank loan, he or she keeps half the loan in cash. Trace out the resulting expansion in the money supply.
Silas puts $1,000 in the bank, of which the bank lends out $1,000 × (1 − rr) = $1,000 × 0.9 = $900. Whoever borrows the $900 will keep $450 in cash and deposit $450 in a bank. The bank will lend out $450 × 0.9 = $405. Whoever borrows the $405 will keep $202.50 in cash and deposit $202.50 in a bank. The bank will lend out $202.50 × 0.9 = $182.25, and so on. Overall, this leads to an increase in deposits of $1,000 + $450 + $202.50 + . . . But it decreases the amount of currency in circulation: the amount of cash is reduced by the $1,000 Silas puts into the bank. This is offset, but not fully, by the amount of cash held by each borrower. The amount of currency in circulation therefore changes by −$1,000 + $450 + $202.50 + . . . The money supply therefore increases by the sum of the increase in deposits and the change in currency in circulation, which is $1,000 − $1,000 + $450 + $450 + $202.50 + $202.50 + . . . and so on.