[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 is

dY/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 it

delta- (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 equation

Y/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 get

k*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, is

Q2 = 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 get

Q2 - 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-t

Taking 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, then

dS/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 read

dS/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 at

dA/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.