[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.

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from my symbols to more conventional ones for the same quantities.