[From Bill Powers (2006.12.20,1625 MST)]

Martin Taylor 2006.12.20.17.58 –

Bill, maybe this whole episode

might have been avoided had I written:p + Gp - Gr = d

instead of

d = p + Gp - Gr.

See my latest (today) post to Bruce. Another reason I’m being prolix here

is that I’m snowed in without a car, am in a cozy warm apartment looking

out at a bleak landscape with snow going by horizontally, and would just

as soon be typing as doing anything else.

What I’m most used to, because of years of modeling, is solving equations

for single variables on the left as functions of the other variables on

the right. Too bad it took me so long to think of the word

“function.” Of course you could reverse the convention and put

the single variable on the right, as you do above, but that wouldn’t

change anything.

Perhaps the core of this matter as far as I am concerned is the fact that

in the expression p + Gp - Gr, p and r cannot vary independently if the

original model is to remain unchanged. I’m sure you can see why this is

so. On the other hand, when we solve for the dependent variable p, we

get

p = [G/(1+G)]r + d/(1+G)

with the one dependent variable on the left, and the two independent

variables on the right. This means we can evaluate the expression on the

right for any pair of values of r and d, and obtain a value of p that is

valid without changing the original model. We can also solve the system

of equations for e, qo, and (trivially) qi, with the same result: each

one is a function of d and r only, with none of the other dependent

variables appearing.

When you isolate an *independent* variable on one side of the

equation, you can no longer think of the expression on the other side as

a function, because the arguments of the function are no longer

independent. And obviously, the value of an independent variable does not

depends on any other variable in the system. In the present case, you

have one dependent variable on the other side, along with the other

independent variable, r.

Are you sure that in your animated diagram, each frame does not represent

the state of a *different model?*Best,

Bill P.

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