Suppose that the Fed's inflation target is 2%, potential output growth is 3.5%, and velocity is a function of how much the interest rate differs from 5%: %^V= 0.5 X (i-5). Suppose that a model of the economy suggests that the real interest rate is determined by the equation r= 8.35-%^Y where Y is the level of output, so %^Y is the growth rate of output. Suppose that people expect the Fed to hit its inflation target. A- Calculate the optimal money growth rate needed for the Fed to hit its inflation target in the long run. B- In the short run, if output growth is just 2% for two years and the equation determining the real interest rate changes to r=4.5-%^Y, what money growth rate should the Fed aim for to hit its inflation target in that period? C- If the Fed intead maintained the money growth rate from part A, what is likely to happen to inflation? D- Which policy do you think is better in the short run? Which is better in the long run?

A- To calculate the optimal money growth rate needed for the Fed to hit its inflation target in the long run, we need to use the quantity theory of money equation: MV = PY, where M is the money supply, V is velocity, P is the price level, and Y is the level of output.

In this case, we can rearrange the equation to solve for the money growth rate (%^M):
%^M = %^P + %^Y - %^V

Given the inflation target of 2%, and potential output growth of 3.5%, we can substitute these values into the equation:
%^M = 2 + 3.5 - 0.5 x (i - 5)

Assuming the interest rate (i) is at 5%, we can calculate the optimal money growth rate:
%^M = 2 + 3.5 - 0.5 x (5 - 5)
%^M = 5.5

So, the optimal money growth rate needed for the Fed to hit its inflation target in the long run is 5.5%.

B- In the short run, if output growth is 2% for two years and the equation determining the real interest rate changes to r = 4.5 - %^Y, we need to calculate the new money growth rate needed to reach the inflation target.

Using the same quantity theory of money equation as before:
%^M = %^P + %^Y - %^V

Substituting the given values and the new equation for real interest rate:
%^M = 2 + 2 - 0.5 x (4.5 - %^Y)

Now, we need to solve for %^M:
%^M = 4 + 2 - 0.5 x (4.5 - 2)
%^M = 5.75

So, the money growth rate the Fed should aim for in the short run to hit its inflation target during that period is 5.75%.

C- If the Fed instead maintained the money growth rate from part A (5.5%), we can expect a discrepancy between the expected and actual inflation rates. As the model suggests, people expect the Fed to hit its inflation target, but if the money growth rate is lower than needed, inflation will likely be lower than the target.

D- In the short run, it would be better for the Fed to adjust the money growth rate to match the changing conditions and keep inflation at the target level. In this case, the money growth rate should be 5.75% to align with the new equation for real interest rate and output growth.

In the long run, the optimal money growth rate needs to reflect potential output growth and velocity. So, it would be better for the Fed to stick with the original optimization and maintain a money growth rate of 5.5% to hit its inflation target consistently.