The MPC and the Multipliers
First: the Spending Multiplier
(either investment spending or government spending)
Y = [ ? ] 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
investment
(I) causes
Noteincreased
that the increase
in income
(Y)
the
economy
spiraltwice
upward
to a new
appears
to betoabout
the increase
equilibrium,
in investmentwhere
(I). the level of income
is higher by Y.
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.
Note
Because
that of
in this
the lessened
economy,attenuation
the increase
of
theincome
successive
rounds of
and
in
(Y) appears
to earning
be several
times
spending,
the increase
the smallinI
investment
drives income
(I).
up by a substantial Y.
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 + bY + I + I
Now, how do you find the difference
between Equilibrium 1 and Equilibrium 2?
Eq. 2.: Y + Y = a + bY + bY + I + I
Eq. 1.: Y
= a + bY
Y =
+I
bY
Y - bY = I
(1 – b )Y = I
Y = [ 1/(1 – b )] I
+ 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.
100.00
80.00
64.00
51.20
40.96
32.77
26.21
20.97
16.78
13.42
10.74
8.59
6.87
5.50
100.00
180.00
244.00
295.20
336.16
368.93
395.14
416.11
432.89
446.31
457.05
465.64
472.51
478.01
Y = 1/(1-b) I
1/(1-b) = 1/(1-0.80) = 5
I = 100
Y = 5 (100) = 500
The MPC and the Multipliers
Second: 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 + G
Suppose 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 + bY - bT -bT + I + G
Now, how do you find the difference
between Equilibrium 1 and Equilibrium 2?
2.: Y + Y = a + bY + bY - bT -bT + I + G
1.: Y
= a + bY
Y =
– bT
bY
Y - bY = -bT
(1 – b)Y = -bT
+I+G
-bT
(1 – b)Y = -bT
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.
80.00
64.00
51.20
40.96
32.77
26.21
20.97
16.78
13.42
10.74
8.59
6.87
5.50
80.00
144.00
195.20
236.16
268.93
295.14
316.11
332.89
346.31
357.05
365.64
372.51
378.01
Y = -b/(1-b) T
-b/(1-b) = -0.80/(1-0.80) = -4
T = -100
Y = -4 (-100) = 400
The Multipliers
Y = 1/(1-b) I
Y = 1/(1-b) G
Y = -b/(1-b) T
}
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The Spending Multipliers
The Policy Multipliers
Suppose we increase G by 100 and increase T by 100.
If b = 0.80, what will the net change in Y?
YG = 1/(1-b) G
YT = -b/(1-b) T
1/(1-b) = 1/(1-0.80) = 5
-b/(1-b) = -0.80/(1-0.80) = -4
G = 100
T = 100
YG = 5 (100) = 500
YT = -4 (100) = -400
Y = YG + YT = 500 -400 = 100
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?
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
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