[From Bill Powers (2000.09.05.0810 MDT)]
Rick Marken (2000.09.04.1420)--
Ok. First, I think B (buying power) must be defined in terms of dollars
paid to the composite consumer. That is, B = PQ-L. That's the way I
define it and it's the way TCP defines it. PQ is, of course, GNP which is the
amount paid to the composite consumer by the composite producer; L is
leakage.
Let's leave leakage out of it at first, just as TCP did. You'll notice that
I'm including two "pools" -- one is inventory, the other is producer
savings. Instead of the output going directly to the consumer, it goes
first into inventory; then inventory is drawn down as the consumer buys
some part of the inventory at the price P. Instead of incoming going
directly to wages and K, they go into savings, and then wages and K are
paid from savings. Neither of these pairs of effects is necessarily in
balance: both savings and inventory can change due to differences between
input and output.
The amount that the consumer buys is B/P -- the amount spent divided by the
price per unit of goods or services. This is not necessarily equal to the
amount produced, Q. If it is less than the amount produced, inventory will
increase; if greater, inventory will decrease. For the time being, we
assume that the consumer spends everything that is earned. It is the job of
the inventory controller, when introduced to the model, to keep inventory
constant (probably close to zero) by adjusting prices. When that control
system is in place, we will indeed find that the price of Q is then such
that the consumer's buying power, without leakage, is just sufficient to
buy all of it. Indeed, (B - L)/P will then equal Q, where at present L is
set to zero.
It is the fact that this model does NOT start by requring that PQ = B
(without leakage) that leaves room for the inventory control system to
operate. And the operation of the inventory control system explains why it
is that at equilibrium, PQ comes to equal B. We are basically explaining
why there is a "circular flow." It is, in part, a consequence of a
purposeful attempt to sell all that is produced, neither more nor less. The
balance is not accidental.
It is the axiomatic requirement that the producer sell all that is produced
that makes this control system a necessary property of the composite
producer. There is no viable business that _consistently_ sells either more
or less than it produces. But neither is it a law of nature that
businesses, even in the aggregate, be viable. In fact, there can be a
difference between the amount sold and the amount produced only as long as
inventory is increasing or decreasing, neither of which is a long-term
option for a composite producer that can survive.
When I used your equation for B I ran into trouble in the equations for
PS and IN. PQ and B had nothing to do with each other. Both are related
to N but the relative size of B and PQ depends on the values I pick for
the constants, w and PR. This seems wrong to me.
Yes, without control systems the whole process will run down or blow up in
some way. That's what we want. The circular flow doesn't just happen: it's
the consequence of the operation of control systems on the basic working,
producing, buying, and selling processes. To see what happens without the
control systems, start the savings and the inventory at some nonzero level
and run the model. Either of these quantities will increase toward infinity
or decrease to zero unless there are control systems at work. You can
adjust wages (or rather N at constant wage) so the total expenditures are
approximately equal to total income, and you can adjust productivity and
prices so the total output Q is equal to B/P, but the slighest change in
any parameter will result in an inevitable drift of Q and B, with
corresponding changes in producer savings and inventory. The system that
stays the same only because of a delicate balance between opposing forces
is not robust against disturbances.
I think the equation for Q is fine but B is really related to how much the
workforce (regardless of its size, N) is paid for producing Q.
B *IS* the the amount that N workers are paid at the yearly rate of w for
producing however many goods are produced (how many are produced depends on
productivity "PR").
The way I have defined B, it is equal to 1.67*w*N, where w is the
(constant) yearly wage per worker and N is the number of workers in the
workforce. The 1.67 multiplier expresses the fact that wages and capital
income are in the ratio (assumed constant) of 60 to 40, so that knowing W
is sufficient to establish K at 2/3 W, so the total cost B = W + K is W +
2/3*W or B = 1.67*n*W.
The cost IS the buying power.
So I define B as
B = PQ-L
As I pointed out above, this creates a fixed relationship between output
and purchases, so inventory and savings _cannot_ change. Approaching this
my way, we allow for B, the amount spent by the producer on wages and
capital income, to be different from (PQ - L), the actual income of the
producer. We should then find that when the two control systems are in
operation, the relationship B = PQ - L _emerges_ (if it actually occurs)
rather than being postulated as a starting point.
The equation for inventory becomes:
IN = IN + (Q - Q')*dt
where Q' is the amount of Q that is actually purchased. Q' is
defined by the following equation
Q' = B/PQ*Q
This is how TCP approached it, but I think it is the wrong approach. He was
trying to anticipate the outcome of all these simultaneous processes; that
is why he carries two different values of output, Q and Q'. But there is
only one actual value of Q, and that is all we need. If we have put the
model together properly, we should find that the output Q will have one
value without leakage, and another value with leakage. Equations that try
to give Q two values at once require, to say the least, a great deal of
explaining. I think TCP was intuitively trying to run a simulation, but he
didn't know how to do it.
I also believe that it makes sense to let inventory, IN, go
below zero. Negative inventory occurs when demand for Q (Q')
exceeds the amount of Q produced. When IN is negative it means
that orders exceed goods on hand.
Let's postpone that consideration (which I think may be useful -- back
orders are tradable commodities). We can also postpone consideration of the
phenomenon of negative savings, otherwise known as borrowing. Both of these
considerations get us into details we're not ready to handle yet, such as
discounts and interest as well as the creation of new money by the
Composite Banker (not yet in the model). Let's see what a more limited
model will do, first.
I have set up the inventory (IN) and cash reserve (PS) control
systems. As you suggested, the inventory manager controls IN
by varying the price of goods (P'); the cash reserve manager
controls PS by varying N (and, thus, Q). The systems seem to work
although the cash reserve manager will eventually lose control
of PS because the workforce will eventually hit minimum.
Yes, and I think this must happen when there is no new money brought into
the system. Note that when the cash reserve manager varies N, this changes
both B and Q: it affects B because the cost of production changes, and Q
because of the factor of productivity, which makes Q depend on N. There are
clearly some other assumptions possible here, which we should eventually
explore.
The
inventory control system also works, though it oscillates quite a
bit, I think due to all the time integrations between output (P')
and input (IN).
These oscillations are probably real. There are two integrators involved,
which makes it possible for phase shifts, by some paths, to reach 180
degrees. To eliminate them I would guess that the inventory controller
should perceive not just the inventory, but inventory plus a "damping
constant" times rate of change of inventory. That will put some
"anticipation" into the loop and help to stabilize it. That may be enough
to stabilize both loops. It may also prove to be highly realistic!
Here is a summary of my revised "plant" equations compared to yours:
Powers Marken
P = producer & consumer cost P = producer cost;P' = consumer cost
Q = goods and services produced Q = goods and services produced
B = 1.67*w*N B = PQ-L
Q' = non-existent Q' = B/PQ*Q; goods and services consumed
PS = PS + (PQ - B)*dt PS = PS + (P'Q' - PQ)*dt
Q = PR*N; Q = PR*N
IN = IN + (Q - B/P)*dt IN = IN + (Q - Q')*dt
The equations describing the inventory and cash reserve control systems
are the same as you [Bill Powers (2000.09.01.1903 MDT)] described them,
though I used integrating output functions.
Note the following on the Marken side (corrections made to be consistent
with programming notation):
B = P*Q - L
Q' = B/P/Q*Q = B/P = (P*Q - L)/P
Typo: there is no P'. You mean PQ', not P'Q' in third from last statement.
Now manipulate the equation for producer savings a little:
PS = PS + (P*Q' - P*Q)*dt, or
PS = PS + P*(Q' - Q)*dt, or
using Q' = (P*Q- L)/P from above,
PS = PS + P*((P*Q - L)/P - Q)*dt, or
PS = PS + P*Q - L - P*Q, or
PS = PS - L
!!!!
We find that producer savings simply decreases at the rate of leakage. If
leakage is zero, producer savings remains constant. All this results from,
in effect, assuming the solution B = PQ - L and making it part of the model.
If we let B = 1.67*w*N, which depends only on the wage level and the number
of workers, and Q equal PR*N, depending only on productivity PR and the
number of workers producing, then we are not assuming a balance between
buying power (less any leakage) and producer income. Producer savings will
rise or fall according to whether producer income is greater or less than
production costs, and inventory will rise or fall according to whether the
workers and capital income recipients have too little money to buy the
whole product, or more than necessary to buy it. That gives the controllers
something to do!
I think this is a pretty good start. I await further comments and/or
suggestions.
I think it's an excellent start. Your reply has led me to understand better
not only your approach (and TCP's) but my own. I didn't realize until now
that the statement B = PQ - L amounts to assuming a solution of the system
equations, which should be left for the simulation to produce if it's
really a solution. And I hadn't worked out the real meaning of my model's
details. Sort of weird, as if my basis for creating the model was different
Best,
Bill P.
···
from my basis for understanding it.