The MPC and the MultipliersFirst: the Spending Multiplier

(either investment spending or government spending)

Y = [ ? ] IY = [ ? ] I

The MPC and the Investment Multiplier

If the investment community increases its spending, incomes and consumption will spiral upward in multiple rounds of earning and spending.

Once the process has played itself out, the economy’s equilibrium income will be higher by some multiple of the initial investment spending.

The 45-degree line represents all possible income-expenditure equilibria: Y = E. (But, of course, there is only one point along that line that corresponds to full employment.)

Consumption behavior is given by a linear equation C = a + bY. In this economy, the slope “b,” also called the marginal propensity to consume, is one-half, or 0.5.

Investment spending is added vertically to consumption spending: C + I is total spending for a wholly private economy.

The economy is settled into an initial equilibrium where Y (measured horizontally) is equal to C+I (measured vertically).

Now suppose that increased optimism in the business community causes investment spending to increase by I .

The increased investment (I) causes the economy to spiral upward to a new equilibrium, where the level of income is higher by Y.

Note that the increase in income (Y) appears to be about twice the increase in investment (I).

A second wholly private economy differs from the first one only in terms of the slopes of their consumption equations. This second economy’s MPC is 0.8.

Notice that with a high MPC, this economy is sensitive to even a small change in investment spending.

Because of the lessened attenuation of the successive rounds of earning and spending, the small I drives income up by a substantial Y.

Note that in this economy, the increase in income (Y) appears to be several times the increase in investment (I).

The consumption equation in this third economy is almost flat. Its MPC of 0.1 means that people spend only one dime out of each additional dollar that they earn.

Only a very substantial increase in investment can have an effect on income comparable to that of the other two economies.

The increase in income (Y) doesn’t appear to be much larger than the increase in investment (I). In the limiting case, where MPC = 0, there is no spiraling at all, and Y = I.

The MPC and the Investment Multiplier

More generally, the multiple that relates Y to I is dependent on the MPC, which is simply “b” in the equation C = a + bY.

We can actually calculate an expression in the form of Y = (some multiplier)I

Y = C + I, where C = a + bY

Eq. 1.: Y = a + bY + I

Suppose I changes by I such that Y changes by Y. The new equilibrium is:

Eq. 2.: Y + Y = a + b(Y + Y) + I + I

Eq. 2.: Y + Y = a + bY + bY + I + I

Now, how do you find the difference between Equilibrium 1 and Equilibrium 2?

Eq. 2.: Y + Y = a + bY + bY + I + I

Eq. 1.: Y = a + bY + I

Y = bY + I

Y - bY = I

(1 – b )Y = I

Y = [ 1/(1 – b )] I

Y = [ 1/(1 – b )] I

1/(1 – b ) is the investment multiplier.

We can say, then, that if investment spending increases by I, then the equilibrium level of income will increase by 1/(1 – b ) times that increase.

Let the MPC be 0.80.

Suppose that investment spending increases by 100.

By how much will income increase?

That is, what Y is implied by a I of 100.

100.0080.0064.0051.2040.9632.7726.2120.9716.7813.4210.74

8.596.875.50

100.00180.00244.00295.20336.16368.93395.14416.11432.89446.31457.05465.64472.51478.01

Y = 1/(1-b) I

1/(1-b) = 1/(1-0.80) = 5

Y = 5 (100) = 500

I = 100

The MPC and the MultipliersSecond: the Tax Multiplier(a head tax, which is a lump-sum tax)

Y = [ ? ] T

How do taxes affect consumption behavior?

For a wholly private economy: C = a + bY

For a mixed economy: C = a + b(Y – T)

“T” is a lump-sum tax, a head tax, a poll tax.“(Y – T)” is after-tax income; it’s take-home pay. Macroeconomists call it “disposable income”.

The MPC and the Tax Multiplier

As with the spending multiplier, the multiple that relates Y to T is dependent on the MPC, which is simply “b” in the equation C = a + b(Y – T).

We can actually calculate an expression in the form of Y = (some multiplier)T.

Y = C + I + G, where C = a + b(Y – T)

1.: Y = a + b(Y – T) + I + G

1.: Y = a + bY – bT + I + GSuppose T changes by T, causing Y to change by Y.

The new equilibrium is:

2.: Y + Y = a + b(Y + Y) - b(T + T) + I + G

2.: Y + Y = a + bY + bY - bT -bT + I + G

Now, how do you find the difference between Equilibrium 1 and Equilibrium 2?

2.: Y + Y = a + bY + bY - bT -bT + I + G

1.: Y = a + bY – bT + I + G

Y = bY -bT

Y - bY = -bT

(1 – b)Y = -bT

(1 – b)Y = -bT

Y = [ -b/(1 – b )] T

-b/(1 – b ) is the Tax Multiplier.

So, that if the tax take increases by T, the equilibrium level of income will increase by -b/(1 – b ) times that increase---which is to say that income will decrease by b/(1 - b) time the increase in taxes.

Compare Y = [ - b/(1 – b ) ] T

with Y = [ 1/(1 – b ) ] G

What’s the difference?

Which is bigger in absolute terms?

Let the MPC be 0.80.

Suppose that taxes are reduced by 100.

By how much will income increase?

That is, what Y is implied by a T of -100.

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8.596.875.50

80.00144.00195.20236.16268.93295.14316.11332.89346.31357.05365.64372.51378.01

Y = -b/(1-b) T

-b/(1-b) = -0.80/(1-0.80) = -4

Y = -4 (-100) = 400

T = -100

Y = 1/(1-b) I

Y = -b/(1-b) T

Y = 1/(1-b) G

The Multipliers

}The Spending Multipliers

}The Policy Multipliers

YT = -b/(1-b) T

-b/(1-b) = -0.80/(1-0.80) = -4

T = 100

YT = -4 (100) = -400

YG = 1/(1-b) G

1/(1-b) = 1/(1-0.80) = 5

G = 100

YG = 5 (100) = 500

Suppose we increase G by 100 and increase T by 100. If b = 0.80, what will the net change in Y?

Y = YG + YT = 500 -400 = 100

When b = 0.95, 1/(1-b) = 20; -b/(1-b) = -19

When b = 0.90, 1/(1-b) = 10; -b/(1-b) = -9

When b = 0.80, 1/(1-b) = 5; -b/(1-b) = -4

When b = 0.75, 1/(1-b) = 4; -b/(1-b) = -3

When b = 0.60, 1/(1-b) = 2.5; -b/(1-b) = -1.5

When b = 0.50, 1/(1-b) = 2; -b/(1-b) = -1

When G = 100 and T = 100, then Y = 100.More generally, when G = T, then Y = G = T.Is this true for all values of b?

The Government Spending Multiplier and the Tax Multiplier are always opposite in sign and always differ by one in absolute terms.

The Balanced Budget Multiplier, then, is one.

Suppose that G and T change together--by (G&T).

Y = (spending mult.) (G&T) + (tax mult.) (G&T)

Y = (G&T)

Y = G = T