Since the GDP is the total annual sum of retail buying
and selling, the players in this model are those who
produce for retail sellers, the retail sellers, and
those who buy from the retail sellers.
The retail buyers will be called "consumers". The
others will be called "producers". Among the producers,
we include the retail sellers, calling them "retail
producers".
To explain the process of stimulation, we make the
following two assumptions about consumers and
producers: (While the numbers used here are chosen
to simplify the arithmetic, they approximate reality.)
Assumption 1.
On average, from their gross income, consumers will
shortly pay:
Ct = 30% to all levels of government for taxes,
leaving the consumer with 70% disposable income.
Please note: since 30% = 30/100 = 0.3,
we will use percentage and decimal values as
equivalents. Thus,
Ct = 0.3 and 1 − Ct = 1 − 0.3 = 0.7 = 70%.
In economic literature, the proportion of after-tax
spending from disposable income is called the “marginal
propensity to consume” or MPC. We will assume that,
on average, MPC = 0.9 = 90%. Thus, consumers will,
on average, shortly pay to retail producers for goods
and services:
Cp = MPC × (1 − Ct)
= 0.9 × (1 − 0.3)
= 0.9 × 0.7
= 0.63
= 63% of their gross income.
The proportion of after-tax savings is called the
“marginal propensity to save” or MPS.
Since MPC + MPS = 100% of disposable income,
MPS = 100% - 90%
= 10%
Therefore, the consumers will save:
Cs = MPS × (1 - Ct)
= 0.1 × (1 - 0.3)
= 0.1 × (0.7)
= 0.07
= 7%
of their gross income. This account for the
consumer's entire gross income.
Assumption 2.
On average, from their gross income, producers will shortly
pay:
Pt = 2% to all levels of government for taxes.
Pp = 50% to other producers
for needed goods and services
(e.g., material suppliers, contractors,
utilities, etc.);
Pc = 47% to employees and owners (who are also consumers).
The payment includes employee benefits and
dividends.
That allocates 99% of their gross revenue. Therefore,
producers will, on average, retain
Pe = the remaining 1% of their gross income as
owner’s equity.
Remarks on the Assumptions
We have not made an allocation for investments because
these are made from savings, not income.
In this model, we ignore foreign trade in the belief
that, in the short-term, changes in trade imbalance
would not affect the efficacy of a stimulus. For the
same reasonn, we also ignore short-term inflation
effects.
We remind the reader that we are using averages to
represent frequency distributions depicting the
economic behavior of more than a hundred million
households and businesses.
Ideally, we would take samples of the population
and collect the needed data to form these
distributions. Then we would use a random number
generator to sample the distributions and do a
computer simulation of the economy over several
years. The result would then be a more accurate
picture of the stimulated growth of GDP and the
value of the multiplier than with our use of
averages.
However, in this course, we merely wish to portray
the process of stimulation and arrive at a
reasonable estimate of the multiplier value. We
believe the use of averages for that purpose will
not be misleading.
Follow the Money
To study stimulation, we assume that “Jane Doe”
is a producer who has a government contract to work
on infrastructure. She has received a check for
STIM dollars (STIM = $1,000) for goods delivered
or services performed.
We will first follow the money as it flows through
the economy and calculate the Total Consumer Spending
(TCS) provoked by STIM. From the value of TCS, we
will calculate the multiplier value from equation (5)
of the previous lecture:
M = 1 + (Consumer Spending / Stimulus)
= 1 + (TCS / STIM)
In the (identical) tables below, Column 1 is labeled
“Pp”. The Row 1, Column 1 cell displays the value
of Jane’s stimulus check. The other cells in Column 1
represent producers supplying goods and services. Their
cells display the payments received from producers
represented by the cell immediately above.
Column
1
2
3
Row
Pp=0.5
Pc=0.47
Cp=0.63
1
$1,000.00
$470.00
$296.10
2
500.00
235.00
148.05
3
250.00
117.50
74.025
-
-------
-------
-------
_
_______
_______
_______
Total
$2,000.00
$940.00
$592.20
.
The Row 2, Column 1 cell displays the value of goods and
services that Jane buys from other producers. As required
by Assumption (2), Jane pays 50% (Pp) of her gross
revenue to these producers. That cell and all following
cells of Column 1 display amounts equal to 50% of the
amounts in the cell above.
As a consequence of this procedure, the ratio of the value
of any Column 1 cell (except the first) to the value of the
preceding cell is 0.5 (Pp).
Column
1
2
3
Row
Pp=0.5
Pc=0.47
Cp=0.63
1
$1,000.00
$470.00
$296.10
2
500.00
235.00
148.05
3
250.00
117.50
74.025
-
-------
-------
-------
_
_______
_______
_______
Total
$2,000.00
$940.00
$592.20
.
Column 2 is labeled “Pc”. It represents the first wave
of consumers. Each cell in Column 2 displays the payments
that producers represented by the cell in the same row
of Column 1 pay to their employees and to themselves,
individuals who are now taking the role of consumers. As
required by Assumption (2), the amount is 47% (Pc) of the
producers’ gross revenue.
In Column 2, the Row 1 cell represents Jane and her
employees. The Row 2 cell represents producers who sell to
Jane and their employees. And so forth in the following cells.
As a consequence of this procedure, the ratio of the value
of any cell in Column 2 (except the first) to the value of the
preceding cell is also 0.5 (Pp).
Column
1
2
3
Row
Pp=0.5
Pc=0.47
Cp=0.63
1
$1,000.00
$470.00
$296.10
2
500.00
235.00
148.05
3
250.00
117.50
74.025
-
-------
-------
-------
_
_______
_______
_______
Total
$2,000.00
$940.00
$592.20
.
Column 3 is labeled “Cp”. It represents the first wave
of retail producers. Each cell in this column displays
the retail value of goods and services that first-wave
consumers, represented by the cell in the same row of
Column 2, buy from first-wave retail producers. As
required by Assumption (1), the amounts displayed in
cells of Column 3 are 63% (Cp) of the consumers’ gross
income. These amounts are the retail producers’ gross
revenue and are added to the Total Consumer Spending
(TCS).
Column
1
2
3
Row
Pp=0.5
Pc=0.47
Cp=0.63
1
$1,000.00
$470.00
$296.10
2
500.00
235.00
148.05
3
250.00
117.50
74.025
-
-------
-------
-------
_
_______
_______
_______
Total
$2,000.00
$940.00
$592.20
.
The amount of purchases made by Jane and her employees is
displayed in the Row 1, Column 3 cell. We will name this
amount: “Consumer Spending 1” or CS1. The following cells
in Column 3 (CS2, CS3, etc.) display the amounts spent by
the owners and employees of producers represented by the
corresponding cells in Column 2.
As a consequence of this procedure, the ratio of the value
of any cell in Column 3 (except the first) to the value of the
preceding cell is also 0.5 (Pp).
Column
1
2
3
Row
Pp=0.5
Pc=0.47
Cp=0.63
1
$1,000.00
$470.00
$296.10
2
500.00
235.00
148.05
3
250.00
117.50
74.025
-
-------
-------
-------
_
_______
_______
_______
Total
$2,000.00
$940.00
$592.20
Formulas
Since there is a constant ratio of value less than 1.0
between the amounts displayed in adjacent cells of
Column 3, the numbers form a geometric series of terms.
Elsewhere,we prove that the sum S of such a series
is equal to:
(1) S = A / (1− R),
where A is the first term in the series (CS1)
and R is the constant ratio between terms (Pp).
Thus we have TCS1, the Total Consumer Spending
of the first wave of producers and consumers:
TCS1 = CS1 + CS2 + CS3 + CS4 + ...
= CS1 + CS1×Pp + CS1×Pp2 + CS1× Pp3 + ...
= CS1 / (1 − Pp)
= $296.10 / (1 − 0.5)
= $592.20
Column
1
2
3
Row
Pp=0.5
Pc=0.47
Cp=0.63
1
$1,000.00
$470.00
$296.10
2
500.00
235.00
148.05
3
250.00
117.50
74.025
-
-------
-------
-------
_
_______
_______
_______
Total
$2,000.00
$940.00
$592.20
.
Although we show only three rows of an infinite series, the
Column 3 total is the sum of all its cells. (The reader
will notice that the sums for columns 1 and 2 follow the same
rule.)
Thus, the $1,000 check stimulated TCS1 = $592.20 of spending
by the first wave of consumers. That amount was received by
the first wave of retail producers.
The First Wave Consumer Spending Formula
We can now develope a formula for spending by the first
wave of consumers (TCS1).
In the formula for the sum of a geometric series
S = A / (1 − R), (from equation (1) above)
we have:
(2) A = CS1 (The Row 2, Column 3 cell value)
= STIM × Pc × Cp (from Assumptions (1) and 2))
and
R = Pp (The ratio of each Column 3 cell value
to the following cell value.)
Therefore, the formula for the first wave spending is:
(3) TCS1 = A / (1 - R)
= CS1 / (1 − Pp)
= STIM × (Pc × Cp) / (1 − Pp) (from equation (2))
= STIM × (0.47 × 0.63) / (1 − 0.5)
= STIM × 0.2961 / 0.5
= STIM × 0.5922
Dividing equation (3) by STIM yields:
(4) TCS1/STIM = 0.5922
In equations (3) and (4), the value 0.5922 is
dependent only on Assumptions (1) and (2)and is
independent of the value of STIM, Jane‘s stimulus check.In effect, Assumptions (1) and (2) provide us with a
constant, 0.5922, which converts a stimulus into a definite
fraction of the stimulus as the amount of spending by the
first wave of consumers by the first wave of retail
producers.
It may help to think of the above table as a calculator in
which STIM is the input and TCS1/STIM is the output. The
value of the output is entirely dependent upon Assumptions
(1) and (2).
In the next lecture we derive formulas for TCS and M.
Proceed to: Lecture 9.
FINDING THE VALUE OF THE MULTIPLIER
Thanks for your interest.
Marvin Sussman, retired engineer
If you don’t like the world as it is, change it this way!:1. Depress the Shift Key,
sweep the cursor over the following URL,
click, copy, and paste it on email to friends,
and ask them to do the same:
Keynesian Economics 101
The ONLY way out of this mess
Free lectures on-line:
KeynesForum.blogspot.com2. To print and distribute single-sheet “invitations”
to this site, click here
and ask your friends to do the same.
3. To donate toward advertising this website,
click the yellow “Donate” button below.
You will be asked for your PayPal account password.
If you don't have a PayPal account,
you will need a credit card.
No comments:
Post a Comment