As we observed in this chapter, central banks, rather than purposefully setting the level of the money supply, usually set a target level for a short-term interest rate by standing ready to lend or borrow whatever money people wish to hold at that interest rate. (When people need more money for a reason other than a change in the interest rate, the money supply therefore expands, and it contracts when they wish to hold less.)
a. Describe the problems that might arise if a central bank sets monetary policy by holding the market interest rate constant. (First, consider the flexible-price case, and ask yourself if you can find a unique equilibrium price level when the central bank simply gives people all the money they wish to hold at the pegged interest rate. Then consider the sticky-price case.)
b. Does the situation change if the central bank raises the interest rate when prices are high, according to a formula such as $R-R_{0}=a\left(P-P_{0}\right),$ where $a$ is a positive constant and $P_{0}$ a target price level?
c. Suppose the central bank's policy rule is $R-R_{0}=a\left(P-P_{0}\right)+u,$ where $u$ is a random movement in the policy interest rate. In the overshooting model shown in Figure $15-12,$ describe how the economy would adjust to a permanent one-time unexpected fall in the random factor $u$, and say why. You can interpret the fall in $u$ as an interest rate cut by the central bank, and therefore as an expansionary monetary action. Compare your story with the one depicted in Figure $15-13$