[From Bill Powers (2003.02.11.1902 MST)]
Bill Williams (2003.02.11) --
Between the beginning of Keynes' Chapter 6 and the point where he reaches
the conclusion that income equals consumption plus investment, there are
only 11 pages. But they are very dense, and I'm not sure they are entirely
consistent. I hope you will help me out here.
Here is the start I have made toward putting Keynes' ideas into the form of
a working model.
We start with his symbol A. "During any period of time, an entrepreneur
will have sold finished output to consumers or to other entrepreneurs for a
certain sum which we will designate as A." (p. 52).
Also defined is A1, a sum which has been spent "on purchasing finished
output from other entrepreneurs." Then:
"And he will end up with a capital equipment, which term includes both his
stocks of unfinished goods or working capital and his finished goods,
having a value G."
It is not clear what the components of G are. Does G consist only of
unfinished goods and finished output from other entrepreneurs, plus
whatever stock of manufactured goods is on hand? Are A, A1, and G three
different and disjoint quantities? The very next sentence, starting a new
paragraph, is not very helpful, since it introduces a composite quantity
that has not, as far as I can see, been previously mentioned:
"Some part, however, of A + G - A1 will be attributable, not to the
activities of the period in question, but to the capital equipment which he
had at the beginning of the period."
Why do we add A and G and then subtract A1? Does this quantity have a name?
I will come back to this question.
This is the statement I have referred to as his struggle with the concept
of an integral. What he appears to be trying to say is that at the start of
the period, there was a quantity to which has been added something to
produce a new value designated as A + G - A1. "We must, therefore, in
order to arrive at what we mean by the _income_ of the current period,
deduct from A + G - A1 a certain sum, to represent that part of its value
which has been (in some sense) contributed by the equipment inherited from
the previous period. The problem of defining income will be solved as soon
as we have found a satisfactory method for calculating this deduction."
One of the deductions he comes up with is "Factor cost" F, which is "the
amount paid out by the entrepereneur to the other factors of production in
return for their services, which from their point of view is income" [note
the seeds of the idea of circular flow]. In other words, as I interpret
this, factor cost consists of the sum of wages and capital distributions.
Another is the fictitious "user cost" U which is a hypothetical cost that I
will ignore by giving it a value of zero until shown a reason to do
otherwise. Still another is a _supplementary cost_ symbolized V, which is
exemplified by physical deterioration of machinery requiring expenditures
just to keep the machinery working properly, or to replace it when it
becomes obsolescent.
If we start with A + G - A1 and deduct these costs, we end up with a value
(to which Keynes doesn't seem to assign a symbol, so I will use X)
X = A + G - A1 - F - U - V.
This would represent the value of X at the beginning of the period. The
value of X at the end of the period is given as A + G - A1, and the change
in value would be - F - U - V. That, of course, makes no sense if this is
supposed to be a calculation of income: it makes income the sum of three costs.
Since a literal interpretation of Keynes' words leads to nonsense, we must
try to find one that makes sense. Clearly, the terms A and G represent
gains in value of X, A being proceeds from sales, and G being an inventory
of manufactured but not yet sold goods that were produced during this
period. G also includes unfinished goods, but since they are never sold we
can absorb them into the costs of production by an alteration in whatever
measure of productivity or efficiency of labor we use.
To approach the problem of income from the front door instead of the back
porch as Keynes does, what we need to do is define a starting condition,
and then define processes which, occurring at different rates, change the
value X from the starting value Xo to the final value Xt. The difference Xt
- X0 will be the income. The starting value will be the ending value from
the previous period. We will first write the differential equations
describing how the variables change, and then convert them to integral
equations to allow calculating changes over a finite interval.
dA/dt = Sg, where Sg stands for a rate of sale of the goods inventory, all
quantities here and below being converted to dollars per unit time.
d(A1)/dt = Pf, where Pf stands for rate of purchasing finished goods.
dG/dt := N*E, where N stands for number of workers, and E is the Efficiency
of the workplace in goods/worker-day. "Day" means time unit.
dU/dt := 0;
dV/dt := D*Vk, where Vk is the current value of capital equipment and D is
the fraction of that value lost per unit time.
dF/dt := N*(W+K), where N is the size of the workforce, W is the wage per
unit time and K is the capital distribution per unit time paid to the
workforce.
The change in value of X is
dX/dt := dA/dt +dG/dt - d(A1)/dt - dU/dt - dF/dt - dV/dt
Income over the accounting period is therefore
t = T
I = integral[(dX/dt)*dt] - Xo
t = 0
This is a good stopping point. I may well have left out things (beside U)
or got things wrong. Corrections or agreements are welcome, as are changes
Best,
Bill P.
···
from my symbols to more conventional ones for the same quantities.