Assume the following information for a hypothetical economy in year 1: money supply = $400 billion; long-term annual growth of potential GDP = 3 percent; velocity = 4. Assume that the banking system initially has no excess reserves and that the reserve requirement is 10 percent. Also suppose that velocity is constant and that the economy initially is operating at its full-employment real output.

1. What is the level of nominal GDP in year 1?

2. Suppose the Fed adheres to a monetary rule through open-market operations. What amount of U.S. securities will it have to sell to, or buy from, banks or the public between years 1 and 2 to meet its monetary rule?

By Fisher’s equation of quantity theory of money, the product of the economy’s money supply and its velocity of circulation equals the nation’s GDP.

MV = PQ, where M is the money supply, V is the velocity of money, P is the price level, and Q is the number of domestic products. PQ is the nominal GDP.

When M is $400 billion, and V is 4, the nominal GDP in year 1 is:

GDP = $400 x 4

= $1600

Since the annual growth rate of nominal GDP is 3%, the nominal GDP in year 2 is:

Nominal GDP₂= Compound Growth Rate x GDP₁

Nominal GDP₂= 1.03 x $1600

= $1648

By Fisher’s equation:

$$\begin{gathered} M=\frac{PQ}{V} \\ M=\frac{1648}{4} \\ M=412 \end{gathered}$$

The money supply has to be increased by $12 billion (= 412 – 400) in year 2. The Fed will buy the U.S. securities equal to the ratio of money created to the money multiplier to increase the money supply.

As given, the required reserve ratio is 10% (or 0.1), the money multiplier is:

$$\begin{gathered} m=\frac{1}{r} \\ =\frac{1}{0.1} \\ =10 \end{gathered}$$

The amount of U.S. securities purchased by the Fed is:

$$\begin{gathered} USSecurities=\frac{Moneycreated}{Moneymultiplier} \\ =\frac{12}{10}=1.2 \end{gathered}$$

Hence, the Fed will buy U.S. securities worth $1.2 billion.