[From Rick Marken (2002.11.13.2300)]
Bill Powers (2002.11.13.1607 MST)--
Bill Williams UMKC 13 November 2002 9:16 AM CST --
Rick Marken (2002.11.12)
Attached in an attempt to find the analytical solution of the Giffen
equations. Please check for errors.
Thanks. This derivation is helpful. Your final equation is:
Y = ------------------------------
You note that the Giffen effect occurs when Px/Vx>Py/Vy. As Py/Vy increases
(but remains less than Px/Vx) the denominator decreases and, thus, Y
increases. This is the Giffen effect: assuming Vy constant then an increase
in the unit price of Y (Py) produces an increase in consumption of Y.
The equation shows how Y must vary as Py varies in order to keep V (total
value) and B (budget) constant. It doesn't show _how_ a system would do this.
Thus, the equation shows how Px, Py, Vx and Vy will affect X and Y for a
system that _can_ vary X and Y in a way that keeps V and B constant (under
control). So the equation shows, for example, that the Giffen effect will
occur if the value of the two goods (Vx and Vy) are equal. For example, set
Vx=Vy = 1 in the equation above and you get:
Y = ------------------------------
(Px - Py)
So if Py starts at a value less than Px and starts to increase (but staying
<Px), the denominator again decreases and Y increases. So we can get the
Giffen effect when X and Y have equal value relative to the controlled
variable V. So there is no need for one good to be inferior to another in
terms of the its contribution to the value of V in order to see the Giffen
effect; the effect occurs for equally valuable goods. Cost per value that
matters; in this case Py/Vy must be less than Px/Vx, regardless of the actual
cost of the two goods.
What is not obvious from the equation is that it is also unnecessary for one
good to be inferior to the other in the sense of being less preferred. That
is, the Giffen effect can occur even if, with an unlimited budget, a person
will purchase the same amount of X as Y. There is probably a way to show this
mathematically but it was very easy to demonstrate this using the control
model of the Giffen effect. I was able to set up the model so that X and Y
were purchased in equal amounts when there was no budgetary need to do so.
Then, by simply increasing Py the Giffen effect kicked in: purchases of Y
went up and purchases of X went down.
My conclusion is that "inferiority" of goods, either in the sense of one
good having less "value" than another (with "value" defined as in Bill
Powers' equation above) or in the sense of consumption of one good being
preferred to that of another, is neither a necessary nor a sufficient
condition for the Giffen effect.
In the control theory model of the Giffen effect, what is necessary to
produce the effect is the proper connection between higher level systems
(controlling for V and B) and consumption of X and Y. That is, the system has
to be "wired" correctly in order to exhibit the effect. Basically, what this
means is that the system has to be set up so that it will automatically
increase consumption of the less expensive (cost/value) good (Y) when the
price of that good goes up as a means of maintaining control of _both_ V and
B. If the system is not "wired" properly then, when the price of Y is
increased in a way that would break the budget, the system will simply not be
able to control both V and B. An "improperly" wired system will probably end
up controlling just B (since it can't spend more than it has) and lose
control of V.
So the Giffen effect will only be shown by a system that has learned the
_skill_ of controlling both V and B by properly increasing consumption of the
good with a lower price to value ratio when the price of this good increases.
This is likely to be a _learned_ skill. So one would not expect (based on a
control theory analysis) to see the Giffen effect every time the price of a
good with a lower price to value ratio is increased. One would only expect to
see it in a population that had learned how to do this: a chronically poor
population, for example. I would _not_ expect to see the Giffen effect in a
population had recently moved from affluence to poverty. Such a population
has never confronted the "Giffen" situation before; it has always been able
to control V and B by consuming as much X and Y as wanted.
I have read one report of an animal study of the Giffen effect (Battalio, et
al, 1991) and the results were mixed; half the animals showed a clear Giffen
effect and the other half didn't. I think its possible (assuming that the
studies were conducted properly) that this difference was a difference in
skill. Some animals were able to learn to control both V and B by consuming
more of the less expensive commodity as it became more expensive; the others
were not. The authors didn't report the data I would like to see (total
amount of X+Y consumed) but my guess would be that those animals who did not
show the Giffen effect were getting less V (total liquid intake, in this
case, which is X+Y in the experiment) than they wanted while the animals who
did show the Giffen effect were able to control V more effectively.