[From Bill Powers (2004.05.28.1259 MDT)]
Bill Williams 27 May 2004 11:00 PM CST --
First, in the orthodox scheme, and by orthodox I mean the micro
neo-classical analysis of individual consumers and producers it is assumed
that a consumer is the principle of maximization. Then commodities have a
utility function such as Ux = f( C/ ( X 1 x X 1) ) or since this
isn't
all that clearly expressed -- the Utility obtained from the consumption of
commodity X one is a function of some constant divided by X one squared.
So, suppose you have a mix of commodities and want to maximize your utility
given a limited budget. . You can maximize the utility obtained by
adjusting the consumption of commodities so that the marginal utility, or
dU/dX obtained from each commodity divided by the price of the commodity
is
the same for each commodity
Have you transcribed that equation accurately?
Yes.
The function 1/X^2 doesn't have a proper maximum: it goes to infinity at X =
0.
The maximum utility
would occur at zero quantity of X, and it would be infinite.
Strange isn't it?
Of course you say that utility is some function f of this expression, but
it would be awkward to find a form for that function that would produce
an ordinary smooth maximum like a bell-shaped curve. [ X^2 is the old
Fortran notation
for the square of X].
No it doesn't, but that isn't what it needs to do to make the neo-classical
price
theory analysis work. If you want to say that in some sense it works.
In the orthodox analysis the consumer doesn't maximize in regard to any
particular commodity. Rather the consumer maximizes the total utility obtain
from a variety of commodities subject to a budget constraint. To do this
the consumer distributes expenditure so that the increase in utility
from an increment of expenditure for a commodity divided by the price of
the commodity is equal for all commodities purchased.
Now, looking at the functions you might say, but this can't be right because
when the price goes to zero the consumer would want an infinite amount of
the commodity. Obviously this doesn't happen. Oxygen is free, but we don't
consume an infinite amount of oxygen. But, you don't get points for asking
about this in an ordinary price theory course. The orthodox analysis of
economic behavior is a pseudo-science carried out in mathematical terms.
However, it has been the only analytic explanation that has been available
to explain why when the price of a commodity increases people buy less of
the commodity. They have had to resort to some fancy arguments to justify
ignoring the anomaly of the Giffen case, but they have been successful
in convincing even the radical students that they have accounted for the
Giffen case. As Robert Solo said, "If you make it fancy enough, our
graduate students will believe anything."
I was about to say that the function you suggested wouldn't work in the
orthodox system. And, it wouldn't work in the ideological version of the
system because the system is intended to place consumption in a realm that
is beyond criticism. Then it occurred to me that some people have made
the argument for the sort of function you are suggesting. The point you
are making does pop up in some upper level price theory courses as an
interesting but not very important curiosity.
However, it seems to me that your suggested function could serve a purpose
in improving the program I wrote sometime ago of a two commodity demand
analysis. In my program I used the loop gain to represent the intensity
of motive for consuming a good. It might improve the program to insert
the function you suggest to represent the intensity of the motive and
leave the loop gain alone.
I am going to post this for your consideration now rather than continue
with a comment on the rest of your post. But, I get back to it later on.
Bill Williams