Aggregate Expenditure Model: Keynes reformulated the model, so that investment still
depend on the rate of interest, but savings and consumption depend on
disposable income, and the interest rate.  We will start with a discussion of
a simple model, in which Savings depends only on disposable income, and
add the interest rate effects on savings a bit later.
 

Fundamental Psychological Law of Consumption:

1. Consumption depends on income: C = C(DI),
where DI = disposable income

2. When disposable income rises, consumption rises, but by less than
the increase in disposable income.

0 < [D in C/D in DI] < 1

[D in C/D in DI] = [Change in C/Change in DI] = marginal propensity to consume, or MPC.

Marginal Propensity to Consume: The change in consumption spending
brought about by a change in disposable income.

Marginal Propensity to Save:  The change in savings brought about by a change in disposable income.

NOTE:

DI = C + S

Change in  DI =  Change in C + Change in S

1 = [DC/DDI] + [D S/DDI]

1 = MPC + MPS

Suppose we are given the following consumption and saving schedule:

($TRILLIONS)
Disposable Income Consumption Spending Savings
1.0 2.0 -1.0
2.0 2.8 -0.8
3.0 3.6 -0.6
4.0 4.4 -0.4
5.0 5.2 -0.2
6.0 6.0 -0.0
7.0 6.8 0.2
8.0 7.6 0.4
9.0 8.4 0.6
10.0 9.2 0.8
11.0 10.0 1.0
12.0 10.8 1.2
Note, every time DI changes by 1, C changes by 0.8, and S changes by 0.2.

Therefore, the MPC = 0.8, and the MPS = 0.2

Using the consumption schedule, we can write-down the equation for consumption. It will be a straight line with slope 0.8,
since the MPC is constant and equal to 0.8.

But, we need to find the value of C when DI = 0. This is the vertical intercept of the consumption function.

Using the schedule, we see that C falls by 0.8 whenever DI falls by 1.0. So, if DI falls from 1.0 to 0.0, C must fall by 0.8,
and reach a level of 1.2.

So, when DI=0, C = 1.2, according to the table.

The consumption function will be:

C = 1.2 + 0.8*DI

In general, C = a + mpc*DI where a - autonomous consumption spending, or, the kind of consumption spending that does not depend on DI, but on something else.

Note, too, that since there are no taxes or transfers, and since we have only one form of spending, C, then DI = Y or real GDP.

As we can see from the table and the graph, we will reach macroeconomic equilibrium, where AE = Y,

At Point E, DI = 6, C = 6, and S = 0

To solve for this algebraically, we set AE = Y.

AE = Y = DI in this case.

But, AE = C.

Therefore, C = Y is the equilibrium, and that occurs at 6.

In a more complicated model, when we have additional sectors of spending, the model works in a similar fashion.

Suppose we have the following spending data:

Notice that C and S are the same as before, but now we have added a new spending component,

I - autonomous investment spending.

Autonomous Investment is Investment Spending that does not depend on the level of real GDP, or Y.

($TRILLIONS)
Y or Real GDP Consumption Spending Savings Investment Spending
    1.0
    2.0
-1.0
    1.0
    2.0
    2.8
-0.8
    1.0
    3.0
    3.6
-0.6
    1.0
    4.0
    4.4
-0.4
    1.0
    5.0
    5.2
-0.2
    1.0
    6.0
    6.0
-0.0
    1.0
    7.0
    6.8
0.2
    1.0
    8.0
    7.6
0.4
    1.0
    9.0
    8.4
0.6
    1.0
    10.0
    9.2
0.8
    1.0
    11.0
    10.0
1.0
    1.0
    12.0
    10.8
1.2
    1.0

In this case, as before DI=Y, since no taxes or transfers or depreciation.

When I = 1.0, We can find the equilibrium value of Y, that is, Y where Y = AE.

Definition: AE = C + I
Algebraic Model Model with data
C = a0 + MPC*Y C = 1.2 + 0.8Y 
I = I0 I = 1.0
AE=[a0+I0]MPC*Y AE=[2.2]+0.8*Y
Equilibrium Condition: Y = AE

Substitute for AE from the above table:
Algebraic Model Model with Data
Line 1 Y = [a0+I0] + MPC*Y Y = [2.2] + 0.8Y
Line 2 Y*= [a0+I0]/[1-MPC] Y*=[2.2]/[1-0.8]= [2.2]/[0.2]
Define the variable k as the expenditure multiplier,

where k = [Change in Y/Change in autonomous spending].

The expenditure multiplier measures the effect on equilibrium Y of a change in

autonomous spending.

The numerical value of k = 1/[1-MPC] = 1/MPS.

For our problem, k = 1/[1-0.8]

So, we can substitute k for 1/[1-MPC] in Line 2 above:
Algebraic Model Model with data
Line 3 Y* = k*[a0+I0] Y*= 5*[2.2] = $11 TRILLION 
Therefore, with the data given above, we find that our new equilibrium,  with the original C data and the new autonomous investment spending level,
I = 1.0, is at $11 Trillion.     This is Point F in the graph above.

NOTICE:

1. An increase of total autonomous spending of $1 Trillion will move the
equilibrium real GDP from $6 Trillion, where it was when C was the only
form of spending, to $11 Trillion, an increase of $5 Trillion.

2. The value of k above is 5. This means that for every $1 increase in
autonomous spending, equilibrium real GDP will rise by $5.

ANALYSIS OF ADJUSTMENT TO NEW EQUILIBRIUM

Note, once autonomousI rises from zero to $1 Trillion:

At Y = 6, AE = 7, AE > Y implies economy expands

At Y = 12, AE = 11.8 AE < Y implies economy contracts

Y = 11, AE = 10+1 = 11, Y is in equilibrium

At Y = 6, producers sell $1 Trillion from the economies inventory
stock to consumers, and produce $1 Trillion of new capital goods.

Change in capital stock = 1.0 Change in inventorystock = -1.0 implies Actual investment = 0

Planned I = 1.0, Actual I = 0, so Actual I = Actual S, always.

Planned I = Planned S, only in equilibrium

Simplifying the solution of Equilibrium GDP.

We said earlier, in equilibrium Y* = [a0+ I0]/ 1-MPC

Let A = total autonomous spending, k = 1/[1-MPC]

Then, Y* = kA

We can always solve for Y* by finding k, and then multiplying it times total autonomous spending.

Suppose we find that Government Spending G, is set at G = 0.4. What is the new equilibrium value of Y?

We have k = 1/[1-MPC]= 1/[1-0.8] = 1/0.2 = 5

A = a0+I0+G0 = 1.2+1+0.4 = 2.6;

Y* = k*A = 5[2.6]= 13

Note, the equilibrium condition means that For ANY increase in a component of total autonomous spending,
we can solve for Y* by using the following:

DY = kDA

Example: Suppose, using the same data we have used up to now, the economy we have been studying opens its economy to world markets. We find that: Exports = 2.7 Imports = 2.2

Let X-IM = Net Exports = Exports - Imports

Then, X-IM = 2.7-2.2 = 0.5.

Total Autonomous Spending will rise by 0.5.

A = a0+I0+G0+[X0-IM0 ]= 1.2+1+0.4+0.5 = 3.1

To solve for Y*, we can use: Y* = kA = 5[3.1] = 15.5

Or, we can realize that A has risen by 0.5, so Y must rise by k times that amount:

DY = kDA = 5[0.5] = 2.5

Previous Y* was 13, so new Y* = 15.5, or an increase of 2.5

Fiscal Policy: Suppose we have the following data for an economy:

C = 2+0.75*Y

I0 = 1.75

X0-IM0 = 0.25

Y = Real GDP

1. What is the equilibrium for this set of data?

k = 1/[1-0.75] = 1/0.25 = 4

A = a0+I0+[X0-IM0] = 2+1.75+0.2+0.25= 4.2

Y* = k*A = 4[4] = 16

Economists for the the Office of Management and Budget determine that the potential level of GDP is 20, and
unemployment at this point will be 5.5%. They argue that something should be done to move the economy to this level,
to lower unemployment.

Advisors to the president argue that an increase in government spending will move the economyto the potential level quickly
and effectively.

2. How much government spending will be needed to move the economy to Y = 20?

There are many ways to do this problem. Easiest way: Y* = kA, and

DY= kDA

We want Y* to rise from 16 to 20. So, Change in Y = 4

We know k = 4.

Therefore,we must have

DA = DY/k = 4/4 = 1

If we increase G from its current value of 0 to a value of 1, then Y* will rise from 16 to 20.

Other solution:

C = 2+0.75*Y

I0 = 1.75

[X0-IM0 ]= 0.25

G0 = ?

Y = Real GDP = 20 at new equilibrium, and k = 4, as before.

Then, Y* = kA implies 20 = 4A

A must equal 5 to make Y = 20, so A must rise by 1, and that must
be the new level of G.
 

Balanced Budget Spending:

The Congressional Budget Office argues that increasing government spending in the manner
recommended by the Office of Management and Budget will create a budget deficit. They argue that
any increase in G must be met by an equal increase in tax revenue, so that the budget remains in balance.

Autonomous taxes: taxes that are independent of the level of GDP.

[This excludes income and sales taxes. Any type of poll tax; would be an example.]

In part 2, we raised G by 1, and moved the economy to Y =20.

Suppose we raise tax revenue from 0 to 1, to keep the budget in balance,

I.e., Tx-G= 0. 3.

If we raise Tx by 1, given that we have reached Y = 20 with our earlier parameters, what will be the new level of GDP?

We have:

C = 2+0.75*DI

I0 = 1.75

[X0-IM0 ]= 0.25

G0 = 1

Tx = 1

DI = Y - Tx

What is the new equilibrium Y?

Solve: Taxes complicate matters considerably.

We now have something new:

C = 2+0.75*DI = 2 + 0.75[Y-Tx]

Taxes are subtracted from real GDP to determine disposable income.

C depends on disposable income, not Y.

We have been able to ignore this difference up to now, since Tx has been zero until now. We solve in exactly the same way, but remember to substitute for DI = Y-Tx.

Definition: AE = C+I+G+X-IM

In this case, substituting in for the variables:

AE = a0+MPC*[Y-Tx]+ I0+G0+[X0-IM0 ]

= 2+0.75[Y-1]+1.75+1.0+0.25

= 2-0.75+1.75+1.0+0.25+0.75*Y = 4.25+0.75*Y

If we want to determine A, autonomous spending, we would get

A = a0+I0+G0+[X0-IM0 ]+[MPC*Tx] = 4.25

In Equilibrium:

AE = Y, so 4.25+0.75*Y = Y Y = 4.25/[1-0.75] = 4[4.25] = 17

Or, Y*=k*A = 4[4.25] = 17
 

Showing effect of interest rate on saving and consumption spending:

As we said earlier in the course, savings depends on the interest rate to some degree.
To show this, we re-write consumption spending:

 C = a(r) + MPC*DI

where a(r) is consumption spending that is sensitive to
interest rates, not to changes in the yearly value of DI.

Examples: Durable consumer goods; washing machines, cars,
stereos, etc., are types of interest-sensitive consumer spending.

Therefore,  a(r) will be inversely related to the real rate of interest:
D a(r)/D r < 0, so that a rise in r will lead to a drop in a(r).

Example: a(r) = 100 - 2r where r - percentage real return

So, if r = 5, then the value of the real interest rate is 5%

 INVESTMENT

 As we said earlier, as market interest rates fall, investment projects that were not
profitable at higher rates become profitable.

 So, we have I(r) as our investment function, where

 DI(r)/Dr < 0.

 Example: I(r) = 200 - 3r

We can incorporate these interest sensitive consumption and investment functions into our aggregate spending model.

C = a(r) + 0.6*DI                  where a(r) = 100-2r

I(r) = 200 - 3r

G0 = 150

[X0-IM0]= 25

Tx = 50

DI = Y-Tx

 To solve for equilibrium real GDP,  if we follow our usual method, then we need to find total Autonomous spending, and calculate the multiplier.

 Multiplier value = k = 1/[1-0.6] = 2.5

Now, autonomous spending will be:

A(r) = a(r) + I(r) + G0 + [X0-IM0] - MPC*Tx

= 100-2r + 200 - 3r + 150 +25 -[0.6][50]

= 300 +150+25-30-5r

= 445 - 5r

 So, our expression for total Autonomous Spending is

A(r) = 445 - 5r

We cannot determine the exact value of A(r) until we know the   value of r.

Suppose the interest rate is 4%. Then r = 4.

A(4) = 445 - 20 = 425

Using the equation Y* = kA(r), and r = 4

We have: k = 2.5

r = 4

A(4) = 425

so, Y* = 2.5[425] = 1062.5 which will be the equilibrium  value of Y.

Suppose interest rates fall to 2%, so that r = 2.

Then, A(2) = 445 - 10 = 435

The new equilibrium will be: Y* = 2.5[435] = 1087.5

The drop in r leads to an increase in a(r) and I(r), and this   leads to an increase in A(r).
 

Notice: a(r) = 100 -2r

a(4) = 92, a(2) = 96

I(r) = 200 -3r

I(4) = 188, I(2) = 194

How much did A(r) change when r fell from 4 to 2?

From 425 to 435, an increase of 10.

How much did a(r) change when r fell from 4 to 2?

From 92 to 96, an increase of 4

How much did I(r) change when r fell from 4 to 2?

From 188 to 194, a change of 6

 Note: Change in a(r) plus the change in I(r) gives the

change in A(r), 4 +6 = 10.
 
 

Income taxes add another level of complexity to the problem.  We have the following data for the economy.

C = 100+0.9*DI

I0 = 300

G0= 400

[X0-IM0]= 50

Tx= -20+0.15*Y

DI= Y-Tx

In this problem, Tx is a function of Y. This means that  the slope of the AE function will no longer be simply the

MPC. Induced taxes imply that the value of the multiplier  will be affected: It will no longer be k = 1/[1-MPC]
 

The tax function Tx = T0+t*Y where T0 is the autonomous level of taxes and 0<t<1 is the tax rate.

 In our problem, T0 = -20 and t = 0.15

 This is a "flat tax" of 15%, with a deduction level of  20 for the economy.

So, substituting for Tx in the definition of DI, we find:

DI = Y-Tx = Y-[T0 +t*Y] = - T0 +[1-t]*Y

For our data,

DI = 20+0.85*Y

To solve:

AE = C+I+G+X-IM, as always

Substituting:

AE = [100+0.9*DI]+300+400+50

Since we saw that DI =20+0.85*Y, we have

AE = 100+0.9*[20+0.85*Y]+300+400+50

= 850+18+0.765*Y = 868+0.765*Y

In algebra: AE = A + MPC(1-t)*Y

In equilibrium:

AE = Y or Y = A+MPC(1-t)*Y

Y = A/[1-MPC(1-t)] or A/[1-MPC+t*MPC]

We have a new formula for the multiplier with income taxes:

k = 1/[1-MPC(1-t)] = 1/[1-MPC+tMPC]

Note, this value will be smaller than k:

k < k, since 1/[1-MPC(1-t)] < 1/[1-MPC]

In our example, k = 1/0.9 = 10 k = 1/0.235 = 4.25

So, the equilibrium Y can be found by:

Y* = kA = 4.25[868] = 3689

Notice, with no income taxes, the multiplier value would

be 10, not 4.25. With the same level of A, the equilibrium

would have been 8680, instead of 3689.

The presence of an income tax reduces the value of the  expenditure multiplier.