While SFEcon adds nothing to the
established qualitative aspects of production and
utility theory, we are interested in starting an
argument about how a superior quantification of
these mundane issues can advance the impulse of economic science.
Though much indebted to those who
have established the geometric analyses following
from diminishing marginal utility, our program
nonetheless collides with received economic praxis at
a point where this generality has become algebra. And there is no concealing
a sentiment that economics might be well served if
Professors Cobb and Douglas would kindly lean into
the strike zone to take one for the team.
The
General Utility Problem
SFEcon can only achieve its
synthesis by way of a uniquely effective way to
master economics' most fundamental generalization,
viz.: diminishing marginal utility. This generalization is usually
imbibed through courses in Production Theory,
Managerial Economics, or Microeconomics. Such course names are unfortunate
for SFEcon because we find it necessary to sort out
these elementary topics rather differently than economics proper.
Three points are needed to frame a
discussion of what makes the SFEcon models
functionally unique.
1. As a matter of definition only, let
us gather up all the geometry and analyses taught
under the course headings above, and re-categorize
them as treatments of 'the General Utility Problem'
(as it was described when formulated in the
Nineteenth Century). This problem operates
on three vectors of order n+1,
where n is the number of inputs to a
production function.
One vector describes a locus of
technical optima, i.e.: the variable curvature of a
production function with respect to each of its n+1
axes.
Another vector records output and
input rates that are economically optimal with
respect to . . .
A third vector containing prices for the
output and for each input.
Solution to the General Utility
Problem would yield the content of any one of these
vectors if the contents of the other two are known.
2. Let us admit that
economics,
despite its protestations to the contrary, cannot
solve the utility problem in any general way
whatever. The very functional forms economics accepts as describing production tradeoffs,
such as Cobb-Douglas, are intrinsically indifferent
as to whether utility is positive or negative,
diminishing or rising, at the margin. Unless their parameters are
contrived, these functions do not even present a
geometry with which to envision an economic optimum. Even when properly conditioned
production parameters are contrived for quantitative
demonstrations in the classroom, disclosure of the
optimum requires unjustifiable exogenous
specifications (a budget constraint or output level)
and excessive computational technique (Lagrangian
transformations or Newtonian iterations).
3. And let us accept the many as yet
uncontroverted indicia that solving the utility
problem would give us no empirical purchase on the
activities of individual firms in competition with
other firms. This is to say that, whenever an
economist has looked to see if a particular
corporation was operating where marginal revenues
equal marginal costs, he has found that it was not.
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The
Hyperbolic Solution
SFEcon's program requires nothing
less than a closed-form solution to the General (and
Household) Utility Problem(s) on behalf of a specific
functional form that also happens to be empirically
robust. While we have our solution, we have
yet to find a journal editor willing to 'embarrass
himself' by submitting the proof to a jury. Hence our present strategy of
delivering SFEcon as a pedagogical tool, so that
students might invite their professors' consideration
of the following two points.
1. When stated in terms of hyperbolic
production functions, all aspects of the General
Utility Problem are solved in closed-form. This claim can be ridiculed because
it requires a simultaneous solution of n+1
equations in n+1 unknowns, where the equations
themselves are all of order n or
n+1. The production function itself
provides one of these equations, and n
more equations require that each input's marginal
value equal its price. Any possibility of a general
solution to this system is comprehensively denied
because of the 'polynomial factoring problem', i.e.: there is no general way to extract roots for
even a single equation of order greater than four. Absent a method for extracting the
roots of higher-ordered equations, general solutions
to the higher-ordered simultaneous systems describing
economic adjustment are unthinkable.
SFEcon offers no clever
approximation for circumventing this computational challenge. We only offer to prove
unequivocally and directly that a certain
parameterization of the hyperbolic form will disclose
real, positive, and empirically meaningful roots to
the specific, higher-ordered, simultaneous system
that is thought to control economic adjustment -
irrespective of the system's size, order, or
numerical content. The hyperbola simply fits the
elementary computational needs of economic science as
precisely and uniquely as the Schrödinger Equation
fits the Periodic Chart of chemical elements. Applying this unique functional
form to economics' defining problem always allows the
closed-form solution to the General Utility Problem's
system of n+1 equations in n+1
unknowns, even though the system of equations is of
order n+1.
Our proof is posted in a 26-page
monograph. This length is only needed to
explore all facets of the General Utility Problem,
including the computation of absolute values of
marginal product from references to ordinary
financial state variables. A basic solution to the utility
problem occurs in this monograph's first six pages - four and a half of which are required to state the
problem. Establishing the algebraic core of
SFEcon is almost trivial, once the possibility of its
being correct has been allowed. The case simply cannot be put in a
way that does not upset the most settled science in
this matter.
2. While this admittedly difficult
conclusion has been accepted by each of the small
number of economists who have considered its proof,
we might speculate that these results have been
generally concealed from economics by one of the most
religiously self-imposed blind spots in the history
of scientific exploration. It seems that the possibilities of
hyperbolic functions are repulsive for their peculiar
hostility to a primary claim of economic science. Hyperbolic descriptions of
productive tradeoffs and personal indifference would
dictate hyperbolic shapes for schedules of supply and
demand. If this shape were correct, then
all computations of elasticity would result in unity,
nullifying any claim that supply/ demand analysis can
portray even the most obvious behaviors presented by
economic adjustment. SFEcon can accept this conclusion
because supply/ demand relations are entirely
peripheral to its operation - this even though its
models operate toward a general working-out of the
continuously-changing elasticity relations that
everything has with everything else.
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Applicability
to Sectors vs. the Firm
SFEcon
models are characterized by boundary
conditions expressing production
tradeoffs for the industrial sectors and utility
tradeoffs for the household sectors. So we
are, from the outset, using the presumed geometry
of production in an untoward manner, i.e.: to
supply the parameters needed for animating an input/output
matrix. Thus
production theory is not being applied to the
individual firm (where there is scant record of
empirical relevance) but to economic sectors composed
of many rival firms (whose competition might result
in the overall picture of optimality on which the
theory relies).
If production parameters are the
proper identifiers of the economic sectors composing
a macroeconomic model, then these agents must be able
to operate near to their 'microeconomic optima'. But, given that the individual
firms are not able to equate marginal revenues with
marginal costs, how could the sectors composed of
such firms be directed toward an optimum?
Perhaps competition among the firms
composing a given sector will drive the whole sector
toward optimality in order to survive in its
competition for capital with other sectors. This would envision movements on a
sector's composite production function as being
effected mostly by individual firms' entering or
leaving the field, merging together, shutting down or
reviving marginal facilities, etc. Where such behaviors would not be
generally available to the individual firms that have
been the subjects of microeconomic analysis, they are
manifestly among the options available to the
economic sectors composed by the firms.
Investigations on this
proposition's empirical soundness can only proceed if
there exists a functional form with the properties
claimed for the hyperbola. The first test of such a function
would be its computability on behalf of real data
that fully describe a macroeconomic system. This
test is realized in a matrix of the hyperbolic
production parameters for a consolidated BEA
benchmark table that the auditor will find appended
to downloadable hyperbola paper.
A
second test would apply the SFEcon algorithm to various parametric sets of similar composition in order to
demonstrate the dynamic stability of models based on hyperbolic production
functions. These
tests have also been performed using the National
Science Foundation's supercomputers. Though
we have yet to acquire the stature necessary to
compose the hyperbolic parameters underlying the
global economy for any lengthy period, we have
invited students to disprove the
empirical possibilities of hyperbolic production
functions in experiments of their own.
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Anecdotal
Evidence
While SFEcon's algebraic claims on
behalf of the hyperbola have yet to impress
themselves on economics, they have directed informal investigations of the sort that should have by now
revealed empirical deficiencies in the hyperbolic
description of economic sectors. The following (completely
un-refereed) work was submitted as course projects by
MBA candidates studying Managerial Economics.
- Many MBA students are
heirs to a family business that can make
their otherwise private histories of
operating data available for the student's use. Analyses of these data tend
to show that hyperbolic parameters would have
to bounce around in inconsistent ways if
these smallish enterprises were generally
operating at an economic optimum. So application of the
hyperbolic form would tend to confirm prior
findings about Diminishing Marginal Utility's
inability to explain micro-phenomena.
- Other students had
considerable work experience giving them
access to operating data compiled over long
periods for whole industries. These studies tended to
identify consistent production tradeoffs
beneath the aggregated activities all the
firms composing a sector. A treatment of such data
for the Aluminum Industry discovered
optimization around a consistent hyperbolic
production function that persisted across
broad operational changes caused by the
energy crises of the 1970's.
- Published input/output
tables were another ready source of data. Over a course of several
years, classes gradually defined a score of
sectors that would span economic activity for
use in the SFEcon system. They then consolidated 1975
-1977 US, West German, and Japanese I/O
Tables into this unified definition of
sectors, and extracted the hyperbolic
production and utility parameters for the
three large industrial economies. Their efforts indicated
that (for at least one point in time), the
dominant economic powers were using the same
technologies, even while their populations'
utility functions were quite different.
- Production parameters
drawn from series of consolidated BEA tables
suggest that hyperbolic production parameters
are constant over time for many sectors of
the US Economy; that the inconsistencies
reflect changing technique in the right
sectors (such as medicine, information
technologies, and petroleum extraction); and
that established methods (e.g., learning
curves) can anticipate hyperbolic
descriptions of technical change.
- The most suggestive of
these amateur studies was by a student who,
anticipating a venture of his own, had
collected historical operating data for all
the boutique wineries in California. He found that none of these
concerns had a believable production
function, at least as one might be described
by hyperbolic parameters. But this industry's
consolidated production function did validate
the essential premises of 'microeconomic' analysis. The marginal rate of
technical substitution between wine and glass
bottles (comprising 90% of winery costs)
equaled the inverse ratio of their prices
over periods in which these prices moved
independently of one another.
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Dynamics and
the Hyperbola
If Economics went back to the drawing board on
production theory, it could find excellent reasons,
owing to pure mathematical
dynamics, for using
multi-dimensional hyperbolae in describing productive
and household indifference. Hyperbolic functions are seen to
control events wherever patterns of accumulation and development are
in evidence; and there are scarcely any real-world dynamics that
might be formally modeled if this one functional form were absent
from human memory.
- The two things one might wish to
read-off a production function, viz. supply
and demand, would logically be integrated
over time in order to create a market
variable; and references to quanta on the
market would, in their turn, establish the prices
that would have to control any economic
model.
-
- Since the integral under a hyperbola
is a natural logarithm, hyperbolic
expressions of economic tradeoffs would
automatically insert the number e
into economic analysis. This
would also bring in the organic references to
Fibonnaci and Taylor series which seem to
underlie every precise understanding of
dynamic phenomena.
These esoteric themes
are given further development in a separate essay titled 'Why the
Hyperbola?'
In
all, the multi-dimensional hyperbolae, used by SFEcon to
describe productive and household indifference, would seem
to fail economic science only insofar as they are novel. We hope this introduction leads auditors to
our downloadable hyperbola paper. It describes all of the hyperbola's
properties vis-ā-vis economic computations, and sets
out the nomenclature used to describe SFEcon's
essential expressions of cause and effect.
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