[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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