Chapter 13 Appendix: Problems 
2. Suppose the required reserve ratio is 10 percent. Assume that the banking system has $20 million in deposits and $5 million in reserves. Find the required reserves, excess reserves, and the maximum amount by which demand deposits could expand. (The capital A in this problem stands for the delta triangle symbol)
The required reserve ratio gives the percent of deposits that banks must hold as reserves. It is the ratio of required reserves to deposits. If the required reserve ratio is 10 percent this means that banks must hold 10 percent of their deposits as required reserves. If deposits are $20 million, then $2 million ($20 million x .10) must be held as required reserves.

Excess reserves are reserves over and above required reserves. If total reserves are $5 million and required reserves are $2 million, then excess reserves are $3 million ($5 million less $2 million).

If the banking system were to loan out its entire excess reserves, the money supply would expand initially by $3 million. However, as this money circulates through the system, there would be further increases in the money supply. The maximum amount by which demand deposits can expand is given by the equation:

ADD = AER/r.

ADD is the expansion of demand deposits, AER is the excess reserves in the banking system, and r is the required reserve ratio. Thus, the maximum amount by which demand deposits can expand is equal to $30 million ($3/0.10).

3. Suppose the required reserve ratio is 25 percent. Assume that banks lend aH of their excess reserves and the public does not add to its cash holdings. Briefly explain how the Federal Reserve's purchase of a $1,000 U.S. Treasury security will affect the money supply. Suppose the Federal Reserve had instead sold a $1,000 security. How will this action affect the money supply? (The capital A in this problem stands for the delta triangle symbol.)

If the Federal Reserve purchases a $1,000 U.S. Treasury security from, say, Jonathan, it will pay for the purchase with a check drawn on a Federal Reserve bank. If Jonathan deposits this check in his bank, Central Security, the bank's demand deposits will increase by $1,000. Because demand deposits are one component of the money supply, the increase in demand deposits results in an increase in the money supply.

Further increases in the money supply will also occur. When Jonathan's check dears, the Federal Reserve will credit Central Security's deposits at the Federal Reserve bank with $1,000. Central Security's reserves will increase by $1,000. Suppose the required reserve ratio is 25 percent. With an increase in reserves of 25 percent Central Security must increase its required reserves by $250 ($1,000 x .25). Its excess reserves will increase by $750 ($1,000 less $250). By assumption, Central Security will loan out these excess reserves.

Suppose the bank lends the $750 to Sallyann who buys a stereo from Mega Sounds. Assume Sallyann's check is deposited in Mega Sounds' bank, Hugo First. Mega Sounds' demand deposits, and hence the money supply, will increase by $750. In turn Hugo First's reserves will go up by $750. It must hold $187.50 as required reserves ($750 x .25) and will now have $562.50 ($750 less $187.50) in excess reserves that it can lend.

The money supply will continue to increase until there are no excess reserves left in the banking system. This occurs when the initial increase in reserves of $1,000 has be absorbed as required reserves. Under the assumptions given, an equation showing the maximum expansion of demand deposits (and hence the money supply) can be derived. It has just been stated that the money supply will increase until the initial increase in reserves becomes required reserves. This can be represented by:

AR = ARR.

AR in this case is the initial change in reserves, and ARR is the change in required reserves that must occur.

We know that required reserves are equal to the bank's reserves multiplied by the required reserve ratio. Further, we know that in this simplified example the bank's reserves are equal to its demand deposits. This means that in the above equation ARR is given by the change in demand deposits multiplied by the required reserve ratio. This is represented as:

r x ADD.

r is the required reserve ratio, and ADD is the change in demand deposits that initially occurred when the Federal Reserve purchased the security.

Substituting the expression for ARR into the original equation gives:

AR = r x ADD.

If both sides of the above equation are divided by the required reserve ratio, r, it becomes:

ADD = AR/r.

This is the equation that shows the maximum amount by which demand deposits can expand. In our example AR is $1000 and r is 25 percent. Substituting these numbers into the above formula we find that the maximum amount by which demand deposits, and hence the money supply, can expand is $4,000 ($1,000/.25).

If the Federal Reserve had sold a security instead of purchasing a security the money supply would have decreased. For example, suppose instead of purchasing a $1,000 security from Jonathan the Federal Reserve had sold him a $1,000 security. Jonathan would pay for the security with a check drawn on his bank, Central Security. This would cause the bank's demand deposits, and hence the money supply, to fall by $1,000. Further, as the check cleared, the Federal Reserve would decrease Central Security's reserves by $1,000. This would decrease its required reserves by $250 ($1,000 x .25). It would also decrease the bank's excess reserves by $750 ($1,000 less $250). This decrease in excess reserves would curtail the bank's lending activities thereby leading to successive decreases in demand deposits and therefore the money supply.

For example, suppose Central Security called in $750 loan owed by Jeannie. Assume that in order to pay the loan Jeannie writes a check from her account at Hugo First. This would cause Hugo First's demand deposits (and the money supply) to fall by $750. Once the check cleared, the Federal Reserve would decrease Hugo First's reserves by $750. Its required reserves would fall by $187.50 ($750 x .25) and its excess reserves would fall by $562.50 ($750 less $187.50). This fall in excess reserves means Hugo First must also curtail its lending activities. The maximum amount by which the money supply can fall is given by the previous equation

ADD = AR/r.
