Lecture 11. INDEBTEDNESS: THE STIMULUS EFFECT

The Multiplier Controversy
In previous lectures, we derived a multiplier formula and 
found that, with nominal values for economic factors, the 
resulting multiplier had a value ranging from about 2.5 to 
3.0.  Whatever we found and no matter how reasonable our 
assumptions and calculations may appear to be, the reader
should know that these multiplier values are not widely 
accepted.

We know that the CBO sets the maximum multiplier value to 
2.5 and that Obama's economists won't go above 1.6, but 
even these figures are not accepted by other economists. 
Some objections are purely ideological. The serious 
objections, such as this, are quite technical. In this course,
we will simply ignore the controversy.

For the following argument, we will also ignore the 
results of the previous lectures. Instead we will ask an 
entirely different question: How large does the multiplier
have to be so that the stimulus pays for itself? That 
would be equivalent to asking: How big does the multiplier
have to be to reduce the DR when we ignore all other
factors in the economy?
How Big is Big Enough? 
Bigger is always better. But a multiplier is big enough
when it effectively reduces the DR and sends the economy 
off in the right direction, no matter how slight the 
reduction. Indeed our purpose in this lecture is to 
demonstrate that, depending upon the value of the DR, 
multiplier values that are much less than our calculated 
values can reduce the DR.

Please note: in the following discussion, we will 
ignore other factors in the economy and consider the DR to
be affected only by the stimulus.

Let S be the value of a stimulus,
let M be the effective value of the multiplier,
and let TB be the total tax burden.

Then, adding the stimulus to the ND and subtracting
the revenue growth from the ND :

(1) Post-stimulus ND
 = ND + S - ΔTR
 = ND + S - (M × S × TB).

Adding the GDP growth to the GDP:

(2) Post-stimulus GDP
 = GDP + ΔGDP.
 = GDP + (M × S / 5).

And

(3) Post-stimulus DR

 = [ND + S - (M × S × TB)] / [GDP + (M × S / 5)].

In these equations, the ND, the TB, and the GDP are
constant values set by long-term history while S is a
constant value set by Congress.  The only
variable quantity is M.

If M is too small (i.e., the stimulus fails to provoke 
enough consumer spending), the post-stimulus DR will
(unfortunately) exceed the pre-stimulus value.  So,
we wish to determine the range of values of M that will
increase the denominator in (3) enough to reduce the DR.

To that end, we will derive the value of the break-
even multiplier, M*, for which the multiplier has no 
effect upon the DR.  At that value the pre-stimulus ratio
is equal to the post-stimulus ratio.

Stated algebraically:

(4) ND / GDP = [ND + S - (M* × S × TB)] / [GDP + (M* × S / 5)]

Elsewhere we prove that equation (4) is equivalent to:
              
(5) M* = 1 / [(DR/5) + TB)].

This is the formula for the theoretical break-
even multiplier, M*.  Any value of M greater than M* 
would increase the formula (3) denominator enough to 
successfully reduce the post-stimulus DR below the pre-
stimulus ratio.
Numerical Example
In a hypothetical economy with GDP = $10T and ND = $10T,
the DR = 100%. With TB = 30%,

M*  = 1 / [(DR/5) + TB]
    = 1 / [(1.0/5)  + 0.3]
    = 1 / [(0.2 + 0.3)
    = 1 / 0.5
    = 2.0


Assume that a stimulus S = $1T
 and that the effective multiplier M = 2.01,
 which is barely more than M*
 and 80% of the 2.45 (at MPC = 90%) value  which we
 calculated in Lecture 9.

Then Post-stimulus ND
 = ND + S - ΔTR
 = ND + S - (M × S × TB)
 = $10T + $1T - (2.01 × $1T × 0.3)
 = $10T + $1T - $0.603T
 = $10.397T

Post-stimulus GDP
 = GDP + ΔGDP
 = GDP + (M × S / 5)
 = $10T + (2.01 × $1T / 5)
 = $10T + $0.402T
 = $10.402T

Post-stimulus DR
 = $10.397T / $10.402T
 = 99.952%,

which is 0.048% less than the pre-stimulus value 100%.  QED.

Thus, in this numerical example, although the $1T 
stimulus increased the ND and the multiplier was only 
slightly above M* and had an effect much smaller than that
expected by the CBO, the stimulus still reduced our 
indebtedness, pleased our bond-holders, put millions of 
people to work, and added $1T of infrastructure to our 
national wealth.

Since, by definition,
M = 1 + (Total Consumer Spending / Stimulus)
  = 1 + (TCS / S)

Therefore, M - 1 = TCS / S

or, multiplying by S: 

TCS = (M - 1) × S
    = (2.01 - 1) × $1T
    = $1.01T

The $1T stimulus had to stimulate only $1.01T,
of consumer spending to decrease our current DR.  The CBO
expected as much as $1.5T of consumer spending, almost 50%
more than the break even amount. Therefore, in the 
current economy, a $1T stimulus would certainly reduce the
DR even when the ND is increased.  That is the way we 
outgrow our debt, even with deficits.

This is the effect of the DR upon the size of the 
multiplier needed to reduce the DR.

Some economists claim that the Keynesian multiplier is 
negligible or below 1.0 or even negative.  But no 
producers and few individuals put their money in a 
mattress.  Neither buildings nor machines nor mines nor 
boats receive paychecks. Ultimately, all of the money 
flows to individuals who either pay taxes, spend it, or 
save it.

Proceed to:
 Lecture 12. THE BREAK-EVEN 
MULTIPLIER
Thanks for your interest.
  Marvin Sussman, retired engineer

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