The Fed uses many tools to influence economic conditions but their three most common include: 1. Open-Market Operations (OMOs) - the purchase and sale of U.S. government securities. 2. Reserve Requirements (RR): affects how much money banks can create by making loans. 3. The Discount Rate - The interest rate on loans the Fed makes to banks.

Here are annual values for M2 and for nominal GDP (all figures are in billions of dollars) for the mid-1990s.

 Year          M2        Nominal GDP

1993     3,482.0         $6,657.4

1994     3,498.1         $7,072.2

1995     3,642.1         $7,397.7

1996.    3,820.5         $7,816.9

1997     4,034.1         $8,304.3

1. Compute the velocity for each year.

According to the equation of exchange in the textbook, the relationship between money supply, velocity, and nominal GDP can express like the equation below:

 Suppose that the M=money supply=M2, V=velocity, then

MV = nominal GDP

V = (nominal GDP)/M 

So, the velocity each year on the list will be:

Year.                   Velocity

1993                    6,657.4/3,482.0 = 1.912

1994.                  7,072.2/3,498.1 = 2.022

1995                   7,397.7/3,642.1 = 2.031

1996                   7,816.9/3,820.5 = 2.046

1997                   8,304.3/4,034.1 = 2.059

2. Compute the fraction of nominal GDP that was being held as money.

According to the textbook, the equation of exchange can express the demand for money as a percentage, given by 1/V, of nominal GDP. With a velocity of V, for example, people wish to hold a quantity of money equal to 1/V of nominal GDP.

So, the velocity each year on the list will be:

Year.                   Being Held

1993                    (6,657.4)*(1/1.912)=3481.90

1994.                   (7,072.2)*(1/2.022)=3497.63

1995                    (7,397.7)*(1/2.031)=3642.39

1996                    (7,816.9)*(1/2.046)=3820.58

1997                    (8,304.3)*(1/2.059)=4033.17

3. What is your conclusion about the stability of velocity in this five-year period?

In this case, during the five-year period, the stability of velocity might due to the stability of the interest rate. People do not like to hold something that expected to lose their value. The interest rate will affect the extra money that people can earn from deposit it more. The reason for the stability of velocity is likely to be a stability of interest rate policy.

Another reason for this case is possibly the expectation of stability. The expectation of deflation or inflation will affect the money people want to hold, they can turn their money back really fast these days. 

Here are annual values for M2 and for nominal GDP (all figures are in billions of dollars) for the mid-2000s.

Year.        M2                Nominal GDP

2003        6,055.5                $10,960.8

2004        6,400.7                $11,685.9

2005        6,659.7                $12,421.9

2006        7,012.3                $13,178.4

2007        7,404.3                $13,807.5

1. Compute the velocity for each year.

Suppose that the M=money supply=M2, V=velocity, then

MV = nominal GDP

V = (nominal GDP)/M 

So, the velocity each year on the list will be:

Year.                           Velocity

2003                            10,960.8/6,055.5 = 1.810

2004.                           11,685.9/6,400.7 = 1.826

2005                            12,421.9/6,659.7 = 1.865

2006                            13,178.4/7,012.3 = 1.879

2007                            13,807.5/7,404.3 = 1.864

2. Compute the fraction of nominal GDP that was being held as money.

Year.                      Being Held

2003                       (10,960.8)*(1/1.810)=6055.69

2004.                      (11,685.9)*(1/1.826)=6399.73

2005                       (12,421.9)*(1/1.865)=6660.54

2006                       (13,178.4)*(1/1.879)=7013.51

2007                       (13,807.5)*(1/1.864)=7407.45

3. What is your conclusion about the stability of velocity in this five-year period?

In this five-year period, people hold more and more money. And in the short run, it is not reasonable to assume that velocity and output are constants. Due to the expectations of the interest rate, people adjust their holding to react to the condition. In this case, people are likely to expect that there are some reasons for them to increase their holdings.

Suppose the velocity of money is constant and potential output grows by 3% per year. By what percentage should the money supply grow in order to achieve the following inflation rate targets?

Suppose that :

%ΔM = the percentage rates of change in the money supply

%ΔP = the percentage rates of change in the price level

%ΔYp = the percentage rates of change in the real GDP(potential output)

%ΔM ≌ %ΔP + %ΔYp

%ΔP ≌ %ΔM - %ΔYp

If the velocity of money is constant and potential output grows by 3% per year

1. 0% 

If the rate of inflation, %ΔP = 0% is our target, then

%0 ≌ %ΔM - 3%

%ΔM = 3%

The money supply should grow 3%, in order to achieve the inflation rate targets 0%

2. 1% 

If the rate of inflation, %ΔP = 1% is our target, then

%1 ≌ %ΔM - 1%

%ΔM = 2%

The money supply should grow 2%, in order to achieve the inflation rate targets 1%

3. 2%

If the rate of inflation, %ΔP = 2% is our target, then

%2 ≌ %ΔM - 2%

%ΔM = 4%

The money supply should grow 4%, in order to achieve the inflation rate targets 2%

About Marginal Propensity to Consume and the Multiplier

What is the marginal propensity to consume when consumption changes from 7 to 6 and disposable income changes from 5 to 3?

According to the material, the ratio of the change in consumption (ΔC) to the change in disposable personal income (ΔYd) is the marginal propensity to consume (MPC). The Greek letter delta (Δ) is used to denote “change in.”

The marginal propensity to consume (MPC)=(ΔC)/(ΔYd)

The change in consumption (ΔC) is from 7 to 6, so the change is -1 (If we focus on the "change", not the direction of positive or negative, that answer is 1)

The change in disposable personal income (ΔYd) is from 5 to 3, the change is -2 (If we focus on the "change", not the direction of positive or negative, that answer is 2)

The marginal propensity to consume (MPC)=(ΔC)/(ΔYd)=(-1)/(-2)=0.5

If disposable personal income is 10 and consumption is 12, what is personal savings? 

According to the material, personal saving is disposable personal income not spent on consumption during a particular period. ( Personal saving = disposable personal income - consumption )

If disposable personal income is 10 and consumption is 12, then the personal saving will be 10 minus 12, that equals minus 2 (10-12=-2)

What does this mean?

In this case, the consumption exceeds disposable personal income, so we get a negative value for saving and the excess must have come from saving accumulated in the past, from selling assets that earned in the past, or even from borrowing.

It also means that consumption choices could be affected by expectations of income and almost all consumption choices could be affected by it over a very long period. 

What is the multiplier when the change in the equilibrium level of real GDP in the aggregate expenditures model is 9, and change in autonomous aggregate expenditures is 3?

Suppose that :

ΔYeq = The change in the equilibrium level of real GDP

ΔA ̄ = The change in autonomous aggregate expenditures

MPC = marginal propensity to consume 

MPS = marginal propensity to save

And, the multiplier is the number by which we multiply an initial change in aggregate demand to get the full amount of the shift in the aggregate demand curve. 

So, the multiplier = ΔYeq/ΔA ̄ 

The relationship between a change in autonomous aggregate expenditures and the change in the equilibrium level of real GDP.

The multiplier = ΔYeq/ΔA ̄ = 9/3 = 3

According to the material, a change in autonomous aggregate expenditures changes equilibrium real GDP by a multiple of the change in autonomous aggregate expenditures. The size of the multiplier depends on the slope of the aggregate expenditures curve. The steeper the aggregate expenditures curve, the larger the multiplier; the flatter the aggregate expenditures curve, the smaller the multiplier.

What is the multiplier when the marginal propensity to save is 1/3?

The multiplier = ΔYeq/ΔA ̄ = 1/(1-MPC) = 1/MPS = 1/(1/3) = 3

What would happen to the marginal propensity to save when a tax cut was enacted causing the multiplier to change to 5?

Suppose that :

MPS = marginal propensity to save

MPC = marginal propensity to consume

If the multiplier = ΔYeq/ΔA ̄ = 1/(1-MPC) = 1/MPS = 5

then, MPS = 1/5 = 0.2 

Reference

https://my.uopeople.edu/pluginfile.php/588647/mod_resource/content/1/TEXT%20macroeconomics-principles-v2.0.pdf