The money multiplier is the reciprocal of the reserve ratio. Under the assumption that banks do not hold excess reserves, the reserve ratio will be equal to the reserve requirement set by the Federal Reserve. For a reserve requirement of 20%, the reserve ratio is 1/5, and the multiplier is therefore 5. When the multiplier is 5, a banking system with $400 in reserves can support 5×$400=$2,000$5\times \mathrm{\$400}=\mathrm{\$2,000}$ in demand deposits.

If the reserve requirement falls from 20% to 10%, the reserve ratio falls from 1/5 to 1/10, and the multiplier rises from 5 to 10. At the lower reserve requirement, the banking system's $400 in reserves supports 10×$400=$4,000$10\times \mathrm{\$400}=\mathrm{\$4,000}$ in demand deposits.

For a given level of reserves, a higher reserve requirement is associated with a smaller money supply. At the higher reserve requirement, banks must hold a larger fraction of their deposits as reserves. This keeps more reserves away from the money creation process (it keeps new loans from being made, which would lead to more deposits, which would lead to more loans, and so on). Therefore, the higher the reserve requirement, the fewer demand deposits are generated in the money creation process from a given change in reserves.

To increase the money supply, the Fed must buy government bonds. In order to pay for the bonds, the Fed creates money. Its purchase of bonds puts the new money in the hands of the public. Assuming that households do not hold cash, the new money will be placed in demand deposits with banks.

At a reserve requirement of 10%, the money multiplier is 10. Therefore, the money supply will grow by 10 times the initial increase in demand deposits from the Fed's open-market purchase. If the Fed buys $20 worth of government bonds, demand deposits and bank reserves will rise by $20. The $20 increase in reserves will support an increase in the money supply of 10×$20=$200$10\times \mathrm{\$20}=\mathrm{\$200}$ as banks lend out the excess reserves generated by the Fed's purchase.

Uncertain economic conditions cause banks to hold some excess reserves, increasing the percentage of deposits held as reserves from 10% to 25% and increasing the reserve ratio from 1/10 to 1/4. The money multiplier falls from 10 to 4—the reciprocal of the new reserve ratio (1/4).

When banks hold additional reserves, the Fed will have to buy more bonds in order to increase the money supply by a given amount. Specifically, an open-market purchase of $50 worth of bonds (rather than $20) is now required to increase the money supply by $200. When the Fed buys $50 in government bonds, demand deposits and bank reserves rise by $50. With the smaller multiplier, the $50 increase in reserves will support an increase in the money supply of 4×$50=$200$4\times \mathrm{\$50}=\mathrm{\$200}$.

For most of this problem, you should have assumed that households hold money only in demand deposits and that banks do not hold excess reserves. A more realistic model of the banking system would relax both of these assumptions. In practice, banks can hold any level of excess reserves that they choose, much as households can hold as much currency as they like. Since the Fed cannot precisely control the level of excess reserves or the fraction of money households wish to hold as currency, it cannot precisely increase or decrease the money supply. However, it can get pretty close by monitoring the behavior of banks and households and adjusting monetary policy to reflect changes in banks' preferences for excess reserves or households' preferences for currency.