Lecture 7. DEFINING THE MULTIPLIER

Among consumers, by far the largest is the federal
government. Every government purchase of goods or services
for $X immediately adds $X to both the ND and the GDP.

Those who sold goods and services to the government will
spend their government checks to manage their enterprise, 
(i.e., pay others for goods and services). Eventually, the 
chain of economic events will lead to retail consumption, 
increasing the GDP beyond government consumption. Thus, 
government purchases stimulate private consumption.

The government's retail purchase of goods and services
from contractors adds directly to the GDP. When contractors
spend government checks for goods and services from other
contractors and vendors, that is not a retail purchase. But
payments to individuals as salaries, benefits, or as
dividends will eventually lead to retail consumer purchases
and thus become an addition to the GDP. 

Assuming a stimulus spread over a 5-year period, its
effect may be analyzed as follows, where both stimulus and
consumer spending are divided by 5:

In algebraic terms:

(1) Post-stimulus GDP = Pre-stimulus GDP
                      + Stimulus
                      + Consumer Spending

Let M be a multiplier such that:

(2) M × Stimulus = Stimulus + Consumer Spending.

Substituting equation (2) into equation (1) yields:

(3) Post-stimulus GDP = Pre-stimulus GDP
                      + (M × Stimulus)

Subtracting Pre-stimulus GDP from both sides of
equation (3) yields:

(4) GDP growth = Post-stimulus GDP − Pre-stimulus GDP,
               = M × Stimulus

Dividing both sides of (2) by Stimulus yields:

(5) M = (Stimulus + Consumer Spending) / Stimulus

      = 1 + (Consumer Spending / Stimulus)

Thus, M is always greater than 1.0.
The Tax Revenue Effect of the Multiplier
When the government and consumers buy retail goods and
services from producers, the amount spent is added to
the GDP. Since tax revenue rises with GDP, the revenue
from the added GDP growth reduces the stimulus cost.

The US tax burden for all levels of government is 
historically about 30% of GDP, which gives us the 
resulting infrastructure purchase discount:

Tax Revenue growth = ΔTR 
                   = 0.3 × GDP growth × 5
                   = 0.3 × M × Stimulus

(This is the total tax revenue increase during
the five-year duration of the stimulus.) 

Discount = ΔTR / Stimulus
         = (0.3 × M × Stimulus) / Stimulus
         = 0.3 × M

Thus, a multiplier value greater than 1.67 would result
in an infrastructure purchase discount greater than 50%.
A multiplier greater than 3.33 would have the stimulus
pay foritself!

Please note: To simplify the explanation in the
calculation above, we have credited the federal
government with 30% of the tax revenue growth
due to the stimulus. In fact, historically, the federal
government gets about 18% of GDP. The states and local
governments get the rest. As a result of this
simplification, our stated values for ND and DR will be
proportionally lower than the correct value.

Our point is
 (1) that the simplification is necessary for proper
     exposition and
 (2) that when the federal government spends, the
     entire nation does get about 30% back in taxes.
     In a fair accounting, the federal government would
     charge the states for the 12% benefit from the
     stimulus.
The Critical Value of M
The value of the multiplier is critical because of the DR. 
Since the tax revenue from the additional GDP will offset 
the cost of the stimulus, we want the multiplier to be as 
much greater than 1.0 as possible. Our purpose in this 
course is to show that, contrary to some opinion, the 
multiplier is substantial enough to reduce the DR.

By definition,

(6) Pre-stimulus DR = ND / GDP.

For a five-year stimulus S and multiplier M,
the average GDP growth,
ΔGDP
 = M × S / 5

And the total five-year tax revenue growth,
ΔTR
 = ΔGDP × TB × 5
 = (M Δ S / 5) × TB × 5
 = M Δ S × TB
 

(7) And the post-stimulus ND
= ND + S - ΔTR
= ND + S - (M × S × TB)

(8) The post-stimulus GDP
= GDP + ΔGDP
= GDP + (M × S / 5)

The post stimulus DR would be the result of dividing 
equation (7) by equation (8):

(9) DR = {ND + S - (M × S × TB)
÷ [GDP + (M × S / 5)]

For the DR to be reduced, Post-stimulus DR must be less 
than Pre-stimulus DR. Algebraically, it must be true that
equation (9) be less than equation (6).

For that inequality to be valid, M must be sufficiently
large.  How large?  That is the central question of
Keynesian/Lernerian economics. It will be discussed in
another lecture.

How Will We Know the Effect of the Stimulus?
Our claim is that a stimulus S will produce a GDP growth 
equal to M × S / 5 during a five-year period but we have
not said why it takes five years to exhaust the effect of
the stimulus. Due to the fact that the federal government
does not have a Department of Infrastructure with a
backlog of “shovel-ready” programs in which to invest,
the entire quantity may be spent over a three-year period. 

In addition, the process of stimulation is a chain reaction 
from vendor to consumer to vendor to consumer, etc. The 
speed of the reaction depends upon many factors including 
the average consumer’s “Marginal Propensity to Consume”
or MPC, described in Lecture 2. During a recession, MPC is 
relatively low. Hence, consumer spending can extend two
years after the last stimulus penny is spent.

And while the stimulated growth is stretched out by both of 
these time constraints, it is also offset by the other 
effects of the recession.  With factory shut-downs, the 
average consumer’s fear increases her “Marginal Propensity 
to Save” or MPS, described in Lecture 2. This causes sales 
to decline, more factory shut-downs, more fear, and more 
GDP decline.

Also, while the federal government increases its spending, 
the state governments are cutting their budgets and laying 
off thousands of employees. Thus, the GDP growth caused by 
the stimulus is simultaneously counteracted by the 
recession’s GDP decline. For all of the reasons given 
above, it will be difficult to determine the effect of a 
stimulus.

The answer to this puzzle is simply to understand that the 
formula:

 GDP growth = ΔGDP = M × S / 5

is correct and needs no post-stimulus confirmation by 
analysis. Regardless of other events, the calculated GDP 
growth will occur over time as S dollars are actually spent
and not hoarded in a bank account.

Given the true value of S, the amount of GDP growth
depends only on the true value of “M”, which
in turn is related only and entirely to the actual process
of producing, spending, and consumption. It is independent
of extraneous factors.

That process is the subject of the next two lectures.
Proceed to: Lecture 8.
THE PRODUCER/CONSUMER MODEL.
Thanks for your interest.
  Marvin Sussman, retired engineer

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