[Fromj Bill Powers (2004.01.21.0745 MST)]
Martin Taylor 2004.01.20.2300 --
, how is it possible to get any kind of dynamical analysis whatever
out of a static base?
I'm sure you know the answer to this, but I'll chime in anyhow.
What you call a "static base" is simply an algebraic analysis. Algebra
knows nothing about time; simultaneous algebraic equations are literally
simultaneous in that all relationships have to hold true at the same
instant. Clearly, then, algebraic equations can't handle any process that
actually takes time to occur, except perhaps to depict the final steady
state if one exists. Keynes tried to handle income, savings, and investment
using algebraic equations, and failed for that reason.
In order to handle processes that take place through time, it is necessary
to use time-dependent expressions, and that means differential equations.
Keynes made a stab at doing this by thinking in terms of accounting
periods. But this doesn't lead to a process-type or time-dependent picture
unless you make the accounting periods shorter and shorter until in the
limit they become zero. In other words, you have to make the transition
from algebra to the calculus.
The way I handled this was to introduce cumulative variables, known in
Vensim as "stocks" and in mathematics as time-integrals. I used a "Reserve"
(to avoid the red-flag word "savings") to indicate accumulations of money,
and "inVentory" to indicate accumulations of goods. Then, as in the post a
week or so ago, one can write the differential equations showing how different
"flow" variables (in Vensim terms) contribute to increases and decreases in
the "stock" variables. Income adds to the Reserve, whether earned or in the
form of distributed producer income, or monetary investments contributed to
a producer. Buying goods subtracts from the Reserve, as would paying
interest or repaying a loan or dropping your wallet into a storm sewer.
Producing goods adds to inventory, and using goods, having them
deteriorate, or selling them subtracts from the inventory. Contrary to
economic custom, I keep track of goods as objects rather than equivalent
monetary value -- one can always convert when necessary by referring to
price per good, summing over goods if different goods have different
prices, and over time as prices change.
Transactions create flows, if treated in terms of transactions per unit
time rather than as isolated events. If you buy at the rate of G goods per
day, for a period of dt days , then your inventory of goods increases at
the rate of G per day, and G*dt per iteration period. The "accounting
period" is dt, which may be 0.001 day. The same transaction rate reduces
your Reserve at the rate of G*P dollars per day, or G*P*dt dollars per
accounting period (P = price).
Now there is no conceptual problem with "instantaneous" movements of goods
and money around the circular flow. The cumulative variables or integrals
decouple the instantaneous changes. They also introduce instabilities that
the real system contains, which are dealt with in various ways (In
Econ004.pas, my first cut at a model, the system comes to equilibrium after
some interesting oscillations). And most of all, they allow simulations in
which all the variables change smoothly through time, which no algebraic
analysis can do.
So, that is my answer to your question, and I'm sure there is nothing new
to you in it.
Best,
Bill P.