[From Bill Williams 15 November 2001 5:00 CST]
[From Bill Powers (2001.11.14.0920 MST)]
"Investment, in the long run, has nothing to do with growth."
If you are going to assert this you'll need some references or an independent
analysis of historical data. And, the "long run" ought to be explicitly defined.
Best
Bill Williams
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[From Bill Powers (2001.11.15.2009 MST)]
Bill Williams 15 November 2001 5:00 CST--
"Investment, in the long run, has nothing to do with growth."
If you are going to assert this you'll need some references or an independent
analysis of historical data. And, the "long run" ought to be explicitly
defined.
I can already think of one way my statement is wrong. Even if investment
doesn't cause growth, it's possible that what does cause growth (such as
advances in technology) might create a market for more investment, so that
without adding money to the system the potential for growth might not be
realized. My hunch is that without an increase in the parameter called
productivity, increasing investment will accomplish nothing. But if
productivity increases, then investment does have something to do with
growth -- its an effect of growth, given that people are always looking for
good investments.
My current guess (get that, GUESS) is that the model will show that there
is no way to increase real investment (money that is not simply diverted
from purchases) other than to create new money. I'm also GUESSING that
without an increase in productivity, an increase in real investement will
just produce inflation. At least that's what I expect the model to show --
not the same thing as being correct with regard to the real economy, of
course.
Best,
Bill P.
[From Bill Powers (2001.11.15.2009 MST)]
Bill Williams 15 November 2001 5:00 CST--
> "Investment, in the long run, has nothing to do with growth."
>
And, the "long run" ought to be explicitly
defined.
Now for the second part. Orthodox economists have introduced time into
there static analysis by way of references to the market day, the short
run and the long run. I would prefer to avoid, because I think this
approach contains inconsistences to work only with a simulataneous
equation model so that:
Y/t = C/t + I/t : The time rate of income equals the time rate of
consumption plus the time rate of Investment.
and
Y/t = C/t + S/t : Substituting time rate of savings
A choice must be made whether to treat I/t or S/t as the independent
variable. I'm choosing to make I/t the independent variable which determines
S/t.
The relationship betweeen a change in investment and the resulting
change in income ( the multiplier 'k' ) can be written as :
k = 1 /( 1 - delta C/t/delta Y/t ) : The change in Investment equals
1 divided by the 1 minus the change in
consumption divided by the change in
income.
Then delta I/t * k = delta Y/t.
THis is the usual notation used to express the relationship between a change
in investment and a change in income. In a sequential analysis the initial
change in I/t is multiplied by C/t/Y/t, if the sequence is continued eventually
the result will be the 1/ ( 1 - delta C/t/deltaY/t ) term. As a description of
economic change howeever this is somewhat clumsy because the equation only
holds for the case in which the sequence is continued infinitely. THe result
has been an extended sequence of arguments concerning "How long is the long
run."
The same relationship can be expressed in simultaneous form as an equation of
time rates thus avoiding the temporal problem involved in the sequential
analysis. So:
dY/t/dt = [I/t - 1/1 - (Ct/Yt - d/C/t/Yt/dt) - Y/t] + [dI/t/dt * 1/1 - C/t/Y/t]
I won't try to express this in english. It's been thirty years since I've
studied the question so I'm more than a bit rusty. However, I think the above
equation correctly specifies the relationships involved. If I had a longer
line I could insert parenthsis to make the equations easier to read, but this
wouldn't change the meaning. If I remember correctly what the above does is
to describe the change in income as being the result of a change in the ratio
of consumption to income ( as time rates ) times the existing rate of
investment as a time rate plus the change in investment times the instaneous
change in the ratio of the time rates of Consumption and Income.
The result is a simultaneous equation of time rates which holds all the time.
THere is no need to bring in short-runs and long-runs and the accompanyin
ambiguities in accounting for change over time.
I may be entirely alone in this, but it is my conclusion that any other
notation for a macro economic theory than a simultaneous equation of time rates
is going to lead at some point to confusions. Whether or not I've correctly
stated the relationship in the above doesn't seem, to me, as important as the
issue of what should be the choice of a fundamental notation.
When I got a copy of Behavior: the Control of Perception one of the first
things I did was to check in the appendix to see how the control theory
equations were specified. Bill Powers used a differential equation notation in
the appendix which as best I could tell handled the time problem correctly.
I've saved a lot of time over the years by checking for this before reading
theoretical work. If the correct treatment is absent, I may quickly skim the
work, but I assume it isn't worth spending much time studying it carefully.
For an extensive and wordy expression of my argument see W.D. Williams M.A.
Thesis University of Denver 1969 "Equlibrium and Equation in Marshall and
Keynes" and Willaims Union Graduate School 1972 "Mathematical Elements in
Veblenian Economics"
Best
Bill williams
______________________________________________________________________
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[From Rick Marken (2001.11.16.0730)]
William Williams wrote:
Now for the second part. Orthodox economists have introduced time into
there static analysis by way of references to the market day, the short
run and the long run. I would prefer to avoid, because I think this
approach contains inconsistences to work only with a simulataneous
equation model so that:Y/t = C/t + I/t : The time rate of income equals the time rate of
consumption plus the time rate of Investment.and
Y/t = C/t + S/t : Substituting time rate of savings
A choice must be made whether to treat I/t or S/t as the independent
variable. I'm choosing to make I/t the independent variable which determines
S/t.
I don't think I understand what you are presenting in this post. Are you
presenting these equations as something to avoid? Or are these proposals for some
basic assumptions of the model?
By the way, if you are talking about rates, then I think you should write the
equations as:
dY/dt = dC/dt+dI/dt
and
dY/dt = dC/dt+dS/dt
Best regards
Rick
--
Richard S. Marken, Ph.D.
The RAND Corporation
PO Box 2138
1700 Main Street
Santa Monica, CA 90407-2138
Tel: 310-393-0411 x7971
Fax: 310-451-7018
E-mail: rmarken@rand.org
[From Bill Williams 16 November 2001 23:00 CST]
For more than a decade Bill Powers has made repeated requests that I join
him in colaborative effort to develop a general model of the market process.
I have declined, because as I have repeated told him, I view Keynes 1936
_General Theory_ as at least a proximately adaquate theoretical treatment
of the macro economy. Powers, however, without bothering to read Keynes
has strong opinions as to what Keynes acomplished-- nothing. This opinion
is derived from Bill Powers dad, an accomplished chemist who didn't find
it neccesary to read Keynes either. I will repeat once again, it is usually
prudent to familiarize oneself with the existing literature before spouting off
about it. ANd, the rewards for rediscovering the wheel are minimual. My
suggestion to Bill would be to find someone more suitable for a fools errand.
[From Bill Powers (2001.11.16.0811 MST)]
[I'm going off-line with this, because this post is a disaster and we have
to decide what to do about it. The"orthodox equations" you present are a
complete sham, and I have to know the extent to which your self-esteem
depends on their validity. If you had known more mathematics and more about
modeling systems you would have seen through the smokescreen, but evidently
you didn't, unless you're holding back on stating your actual opinion of
the orthodox treatment. I deleted a long post when I realized that my
efforts to make sense of the "mathematical" manipulations were futile, and
that to say what I really thought on CSGnet would be a disservice to you.
So basically, I'm starting this post over and sending it directly to you
instead of the net.Last-minute note. Insead of dwelling overlong on what was wrong with the
orthodox approach you described, I will spend some time showing how to
build up a model systematically starting with valid mathematical forms and
manipulations. You're welcome, of course, to correct basic design features,
but be cautious about messing with the treatment of rates and quantities.]
Bill Williams (2001.11.16) --
>Orthodox economists have introduced time into
>there static analysis by way of references to the market day, the short
>run and the long run. I would prefer to avoid, because I think this
>approach contains inconsistences to work only with a simultaneous
>equation model so that:
>
> Y/t = C/ + I/t : The time rate of income equals the time rate of
> consumption plus the time rate of Investment.
>
> and
>
> Y/t = C/t + S/t : Substituting time rate of savings
>
>
> A choice must be made whether to treat I/t or S/t as the independent
>variable. I'm choosing to make I/t the independent variable which determines
>S/t.
---------------------------------------------------------------
Let's look at the above two equations.First, these are not "rate equations". Dividing by t is not how to take a
first time-derivative. What you (they?) should have written isdY/dt = dC/dt + dI/dt, where the "d"s indicate infinitesimal changes.
If you say dI/dt = dS/dt, then it follows, as you said, that
dY/dt = dC/dt + dS/dt.
HOWEVER: You've _assumed_ here that the savings rate is equal to the
investment rate. Since you've assumed this, there is no question of
proving that the assertion is true, and what the model that results does
will depend entirely on the truth of the assumption. I'll try to show you
what one of the implications of this assumption is.If you're going to use this differential-equation notation, then Y, C, and
I would seem to be quantities rather than rates of change. I can see where
this results in confusion, especially for terms such as income, investment
and consumption which seem to be rates in themselves -- perhaps the
"long-term, short-term" confusion to which you refer. How do you determine
the quantity Y so you can measure its change, without seeming to imply a
change in earning rate (i.e., an acceleration)?I'll get back to this. There are worse problems.
> A choice must be made whether to treat I/t or S/t as the independent
>variable. I'm choosing to make I/t the independent variable which determines
>S/t.Yes. You're assuming that which ought to be proven, not assumed. But that's
far from the worst problem.> The relationship betweeen a change in investment and the resulting
>change in income ( the multiplier 'k' ) can be written as :> k = 1 /( 1 - delta C/t/delta Y/t ) : The change in Investment equals
1 divided by the 1 minus the change in
consumption divided by the change in
income.>Then delta I/t * k = delta Y/t.
The notation "delta C/t" etc. is meaningless. Where do you start measuring
t -- at the birth of Christ, or the death of the Buddha, or 8:00 this
morning? And is C the total consumption this year, or since I was born? And
to what does the delta apply? Is itdelta- (C/t) or
(Delta-C)/t?
Or could you possible have meant (delta-C) / (delta-t)?
These would all yield different numbers. But let's go back to
delta I/t*k = delta-Y/t
Assuming that the notation Y/t etc. means the same as in the initial
equations, we have from your first equationY/t = C/t + I/t, and taking deltas of both sides,
delta-Y/t = delta-C/t + delta-I/t.
We can now substitute k*delta-I/t for delta-Y/t, according to the result
you obtained, to getk*delta-I/t = delta-C/t + delta-I/t, or
delta-I/t(k - 1) = delta-C/t
Since the term in C was left unknown after your derivation, we now find
that change in consumptiion has to be a specific function of change of
investment, namely the above relationship if it is true that
delta-investment equals delta-savings. But does it makes sense that
consumption depends on investment alone? This is a nonsensical result, but
it is what the mathematics inescapably implies (whether anyone saw this
implication or not -- not doing the calculation is not the same thing as
disproving the result).Now let's get back to the quantity problem and see if we can't handle it
correctly.Income, Y, is normally thought of in terms of a rate at which money is
earned: dollars per hour, week, month, or year. So strictly speaking, dY/dt
ought to mean the rate at which income is changing -- a positive value
means your wage or salary is increasing. But in the equations above, it's
evident that Y (as well as C, S, and I) is being used as if it represents
a quantity whose size is changing, not a change in a rate of change.
Perhaps this is the confusion you were referring to. If so, you are quite
right in saying that we must use differential equations, but we must also
be very careful to distinguish rates from quantities. I always wondered why
Jay Forester made such a big deal of "stocks and flows", but perhaps this
is the problem he, too, was trying to solve.How do you handle something like income which is clearly a flow of money,
when there are other flows that are simultaneouskly subtracting money, and
when there are also stores or stocks of money? The answer, to anyone
versed in differential equations and engineering modeling, is obvious: you
distinguish rates from cumulative quantities.Let Q equal the quantity of money on hand, measured in dollars (not dollars
per unit time, but just dollars). Suppose I start spending money at a rate
of "s" dollars per unit time. The spending rate is now measured in dollars
per time unit, not just in dollars. So Q is measured in dollars, and s is
measured in dollars per time unit. What is the relationship between Q and s?Suppose we want to know how much has been spent between a starting time t1
and an ending time t2, given that spending is steady at a rate s dollars
per time unit. The problem is just like any rate or speed problem: the
total distance travelled is the average speed in miles per time unit, times
the elapsed time. And the total money spent is the average rate of spending
times the length of time that the spending goes on: S = s*(t2 - t1), where
S is a quantity-variable measured in dollars.If we take S out of some larger quantity Q1, the amount of Q we are left
with, Q2, isQ2 = Q1- S, or
Q2 = Q1 - s*(t2 - t1).
The change in the stored quantity of money is Q2 - Q1, which suggests that
we subtract Q1 from both sides of the equation to getQ2 - Q1 = s*(t2 -t1), and
divide both sides by (t2 - t1) to get
(Q2 - Q1)
------------- = s.
(t2 - t1)We can define delta-Q as (Q2 - Q1) and delta-t as (t2 - t1). Thus
delta-Q
------------ = s
delta-tTaking the time interval to zero by the methods defined in the differential
calculus, we finally otain,-dQ/dt = s
The minus sign comes from defining Q as a positive quantity, and s as a
positiuve rate of spending rather than a negative rate of accumulation. A
positive s means a declining quantity of Q, so Q2 is always less than Q1
when s is positive. Hence we insert a minus sign before dQ/dt and consider
both Q and s to be positive quantities unless a minus sign is used to make
them negative. More conventionally we would write dQ/dt = -s. A negative
spending rate would be a rate of accumulation.So how do we handle Y so as to avoid getting confused between quantity and
rate? I think that "income" is most naturally seen as a rate variable, so
let's let it be a rate variable and use another term, like Savings, for the
corresponding quantity variable. If income is steady at Y dollars per unit
time, and it all goes into savings, thendS/dt = Y
which means that the rate of change of savings in dollars per unit time
equals the income Y in dollars per unit time. Obviously, I'm now changing
some definitions from those used above, rather than go back and change all
the letters. S now means Savings.Now consider Consumption, another rate variable (spending so many dollars
per unit time on acquisition of goods and services). This is a
two-dimensional variable, in that there is a quantity of goods and a price
per good to consider. We can see that if G goods are acquired at a price p
per good, p*G dollars have been spend. If we are acquifring G goods per
unit time, we are spending p*G dollars per unit time. Consumption measured
in dollars per unit time is therefore p*G and it is a rate variable because
p is a scalar, and G is a rate variable.The money being spent comes from the same place where income goes: an
internal store of money which incliudes savings accounts, checking
accounts, and cash -- we'll just call it savings for now. We have already
shown savings increasing at the rate Y, so dS/dt = Y. We now have a rate of
loss of money from savings, so we can revise the differential equation to readdS/dt = Y - G*p, or letting G*P = C,
dS/dt = Y - C.
Now Y and C can be rate variables, while S is a quantity variable.
This leaves I to be accounted for. I is money taken directly from the
consumer's money supply and transferred to the producer as an investment,
without any goods changing hands. If we treat I also as a rate variable, in
dollars per unit time, we find that I drains consumer savings at a rate
equal to the investment rate, so we must make one last revision to the
equation:dS/dt = Y - C - I.
For the consumer we must create another equation to keep track of the
goods. Since G is a rate variable, we need a quantity variable in which to
keep track of the cumulative amount of goods that have changed hands: call
it A for Acquisitions. By reasoning like the above, we arrive atdA/dt = G (recalling that G is a rate variable measured in goods per
unit time)Aside from accounting for depreciation and usage of goods, this is all that
can be done for now with the consumer equations. What we now need are the
producer equations to provide the remainder of the system of equations.
First, we have to take care of the "conservation laws".Let V be a quantity variable indicating total goods on hand in the
producer's output pipeline: "inVentory". Also, let R be a second quantity
variable measured in dollars, representing the producer's monetary
Reserves. We can now take care of conserving money and physical goods:dR/dt = -dS/dt (note minus sign: money entering Reserves subtracts from
consumer Savings by the two routes, consumption and investment. Drains on
reserves are not yet accounted for.)dV/dt = -dA/dt (goods Acquired by the consumer come out of the producers
inVentory. Production that adds to inVentory is not yet accounted for).The producer spends money in three major ways: on wages, on capital
expenses to alter productivity, and on distribution of profits to owners,
lenders, and renters. By definition, these expenditures are the total cost
of production. All the money so spent goes to consumers, and constitutes Y,
consumer income. If you also want a producer income rate variable we could
say Yc and Yp. This money comes out of Reserves; if it comes out at a rate
less than p*G, dR/dt is positive (Reserves are increasing); if the rate is
more than p*G, reserves are decreasing.Notice that we have four quantity variables, A, S, R, and V, and all the
rest are rate variables.The producer can vary strategies concerning the rate at which money is
spent on capital goods and distributions and on wages, and on how much
Reserve and inVentory to maintain. The main variables that can be used by
managers to control R and V are wages and prices. Others, of course, could
be added in a more complex model.The consumer can vary strategies concerning how money is spend on goods and
investments. The means of adjusting these things include decisions about
how many hours to work per unit time (affecting wage-earners' income) and
which expenditures to make on goods and investments.That's about as far as I've gone. As you can see, there is nothing
controversial about any of these relationships, and they are all defined at
the lowest level of abstraction that seems possible or at least feasible.
Also, the mathematical treatment is correct to the best of my knowledge
(check with Richard K. if you have doubts). The step from the equations
developed here to a working model do not look too difficult, although we
will have to make a lot of decisions concerning what we put into the
control systems, the managers and consumers.I think you can scrap all those orthodox equations.
I wish you success in your attempt. However, the mountain you are attempting to
scale has, to the best of my knowledge, already been climbed.
Best
Bill Williams
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