 # 5. The Keynesian Model

### Contents

1. Simple Keynesian Model
2. Aggregate Expenditures
3. Equilibrium
4. Consumption Function
5. Autonomous Spending
6. Autonomous Spending Multiplier
7. Government Fiscal Policy
8. Automatic Stabilizers

### 4. Consumption Function

1. In the Keynesian model, the most important determinant of consumption expenditure is:

a. the interest rate
b. government spending
c. government tax and expenditure policy
d. the current level of personal income
e. investment
g. the stock of consumer wealth
h. expectations about future prices and income
i. the availability of credit and level of credit charges.

Answer: As tempting as some of the answers are the only correct answer is (d). While more recent economic models do consider the interest rate, the stock of consumer wealth, expectations about future prices and income, and the availability of credit and level of credit charges, these variables were either ignored or assumed to be of small significance in the consumption function by Keynes.

### 6. Autonomous Spending Multiplier

1. What effect will a \$30 billion decrease in autonomous investment have on total expenditures if the MPC is 0.9?

Answer: A reduction in autonomous investment expenditures of \$30 billion will lower both total expenditures and national income by a larger amount because of the multiplier effect. National income and total expenditures will decline, in equilibrium, by \$300 billion, or \$30 x multiplier = \$30 • [1/(1-0.9)] = \$30 • (1/0.1) = \$30 x 10.

2. What happens to the level of national income in an economy with an MPC of 0.8 and if autonomous consumption increases by \$20 billion, autonomous investment increases by \$5 billion, autonomous government expenditures decrease by \$ 10 billion, and autonomous net exports increase by \$15 billion?

Answer: This is the same as the previous question with three changes. First, we have four different autonomous expenditure categories changing. The trick is to recognize that the effect of each of these on the economy is identical and you simply add the four to get the total change in autonomous expenditures, or \$20 + \$5 - \$10 + \$15. Second, instead of the decrease of \$30 billion in problem 1 we have an increase of \$30 billion. Third, the marginal propensity to consume is smaller and the multiplier will also be smaller. The total change in national income and total expenditures in equilibrium would be an increase of \$150 billion, or \$30 x multiplier = \$30 x [1/(1-0.8)] = \$30 x (1/0.2) = \$30 x 5.

3. Suppose that the marginal propensity to consume in an economy is 0.75 and that there is a change in government spending that leads to a \$20 billion increase in national income and total expenditures? What was the initial change in government spending that led to the change in national income?

a. increase of \$20 billion
b. increase of \$5 billion
c. decrease by \$5 billion
d. decrease by \$20 billion

Answer: B. Here we work the multiplier equation in reverse, where the total change in income = change in government spending x multiplier. With an MPC of 0.75 the multiplier is 4, or 1/(1-0.75) = 1/0.25 = 4. So \$20 = 4 x change in G, and the change in G = \$5 billion,

You should recognize that these three questions are related. First, we must calculate the autonomous spending multiplier:

 Autonomous Spending Multiplier = 1      1 - MPC

Second, given the multiplier we can calculate the relationship between an initial change in autonomous spending and the total change in national income in equilibrium:

```      Total Change    Autonomous   Initial Change
in National  =  Spending  x in Autonomous
Income       Multiplier      Spending```

Another question that could have been asked is one that gives you the initial change in autonomous spending and the total change in national income in equilibrium and asks you to calculate the multiplier or even the economy's marginal propensity to consume.

### 7. Government Fiscal Policy

1. Government purchases and lump sum taxes are \$500 and \$400, respectively. Investment equals \$200. The autonomous part of consumption is \$100. The marginal propensity to consume is 0.9.

AE = C + I + G + NX

C = 100 + 0.9 • YD
YD = Y - 400
I = 200
G = 500
NX = 0

What is the level of GDP?

Answer: Only trick is to recognize that GDP is equivalent to national income

Step 1. Substitute the equations for the four spending components into the aggregate expenditure equation:

AE = C + I + G + NX
= 100 + 0.9 • YD + 200 + 500 + 0
= 100 + 0.9 • (Y - 400) + 200 + 500
= 800 + 0.9 • Y - 0.9 • 400
= 440 + 0.9 • Y

Step 2. Apply the equilibrium condition, equation (2):

Y = AE

Step 3. Substitute AE from Step 1 into the equilibrium condition in Step 2:

Y = 440 + 0.9 • Y

Step 4. Collect the Y terms on the left hand side and solve for national income, Y.

Y - 0.9 • Y = 440
(1 - 0.9) • Y = 440
Y = 440 / 0.1
Y = \$4,400 = the level of GDP

2. Consider an economy similar to Problem 1, but with an income tax that is one-third of income.

AE = C + I + G + NX

C = 100 + 0.9 • YD
YD = Y - 0.33 • Y - 400
I = 200
G = 500
NX = 0

The government decides to increase spending in order to increase GDP by \$750. How much should government spending increase?

Answer: The solution method is the same as the previous problem. The only difference is that we have added an income tax, 0.33 • Y, to the model.

Step 1. Substitute the equations for the four spending components into the aggregate expenditure equation:

AE = C + I + G + NX
= 100 + 0.9 • YD + 200 + 500 + 0
= 800 + 0.9 • (Y - 0.33 • Y - 400)
= 440 + 0.9 • (Y - 0.33 • Y)
= 440 + 0.9 • (0.67 • Y)

AE= 440 + 0.6 • Y

Step 2. Apply the equilibrium condition, equation (2):

Y = AE

Step 3. Substitute AE from Step 1 into the equilibrium condition in Step 2:

Y = 440 + 0.6 • Y

Step 4. Collect the Y terms on the left hand side and solve for national income, Y.

Y - 0.6 • Y = 440
(1 - 0.6) • Y = 440
Y = (1 / 0.4) • 440
= 2.5 • 440

Here the trick is to recognize that all that is required is the multiplier. The multiplier is 2.5.

If desired change in Y = \$750

\$750 = multiplier • change in G
change in G = \$750 / 2.5 = \$300

3. Suppose we expand our model to take account of the fact that transfer payments, TR, do depend on the level of income, Y. When income is high, transfer payments such as unemployment benefits will fall. Conversely, when income is low, unemployment is high and so are unemployment benefits. We can incorporate this into our model by writing transfer payments as:

TR = TR0 - b • Y
b > 0

Consider the economy is further defined as:

AE = C + I + G + NX
C = C0 + c • YD
I = I0
G = G0
NX = 0

YD = Y + TR - TA
TA (lump sum taxes) = TA0

A. Derive the expression for national income, Y.

Step 1. Substitute the equations for the four spending components into the aggregate expenditure equation:

AE = C + I + G + NX
= C0 + c • YD + I0 + G0
= C0 + c • (Y + TR - TA) + I0 + G0
= C0 + c • (Y + TR0 - b • Y - TA0) + I0 + G0

Step 2. Apply the equilibrium condition, equation (2):

Y = AE

Step 3. Substitute AE from Step 1 into the equilibrium condition in Step 2:

Y = C0 + I0 + G0 + c • TR0 - c • TA0 + c • (Y - b • Y)
= (C0 + I0 + G0 + c • TR0 - c • TA0) + c • (1 - b) • Y

Step 4. Collect the Y terms on the left hand side and solve for national income, Y.

Y - c • (1 - b) • Y = C0 + I0 + G0 + c • TR0 - c • TA0

Y =          1          • (C0 + I0 + G0 + c TR0 - c • TA0)
[1 - c (1 - b)]

B. What is the new multiplier?

Multiplier = 1 / [1 - c (1 - b)]

C. Why is the new multiplier less than the standard one?

Answer: Transfer payments act as automatic stabilizers. As income declines (e.g., economy goes into a recession), transfer payments increase.

4. Let us introduce a foreign sector into the national economy. The following equations describe the economy:

AE = C + I + G + NX
C = 20 + 0.75 • (1 - t) • Y
I = 25
G = 15
NX = exports - imports = 20 - 0.1 • Y

YD = Y + TR - TA
TA (lump sum taxes) = TA0

If the government seeks to maintain a zero trade balance (NX = 0), what proportional income tax rate, t, should be set?

Step 1. Substitute the equations for the four spending components into the aggregate expenditure equation:

AE = 20 + 0.75 • (1 - t) • Y + 25 + 15 + 20 - 0.1 • Y
= 80 + 0.75 • (1 - t) • Y - 0.1 • Y

Step 2. Apply the equilibrium condition, equation (2):

Y = AE

Step 3. Substitute AE from Step 1 into the equilibrium condition in Step 2:

Y = 80 + 0.75 • (1 - t) • Y - 0.1 • Y

Step 4. Collect the Y terms on the left hand side and solve for national income, Y. Seems we have a problem since we were asked to solve for the income tax rate, t, but we don't know the value of national income, Y. The trick is to recognize we can obtain a value for Y from restriction that NX = 0. Where it was given:

NX = 20 - 0.1 • Y
0 = 20 - 0.1 • Y
Y = 200

Substitute the value for Y ( = 200) into the equation from Step 3:

200 = 80 + 0.75 • (1 - t ) • 200 - 0.1 • 200
200 = 80 + 150 - 150 • t - 20
10 = - 150 • t
t = 1/15 = 6.67 percent

### 8. Automatic Stabilizers

1. Which of the following represent an example of an automatic stabilizer?

a. unemployment compensation
b. public works designed to get the economy out of a depression
c. local property taxes
d. inheritance taxes
e. the personal income tax
f. temporary tax increases passed by congress to fight inflation
g. income tax

Answer: A, E and G. When the economy moves into a recession, national income declines and unemployment increases. Total income tax revenue automatically declines and total unemployment benefit payments automatically increase without Congress taking any action. Answers B and F represent discretionary rather than automatic fiscal policies. With discretionary fiscal policy Congress and the President must take some action before the spending or tax revenues can change. Answers C and D are neither discretionary nor automatic. During recessions these tax rates usually don't change.