A new model of perfectly competitive monopolist competition when products and factors are a perfectly differentiated continuum of points. General Equilibrium is found to be a soliton (a continuum of fixed points), not a single fixed point as presently conceived in the literature. Further, the general equilibrium core surface is found to be a tension minimizing minimal surface. An application of mathematical economics and welfare economics to be submitted to Economic Letters. Copyright, September 7, 2019, Richard Anthony Baum, Santa Barbara. California, USA, which copyright will be relinquished upon publication.

1. Richard Anthony Baum Submitted to Economic Letters
Department of Economics, Semi-Retired September 7, 2019
Allan Hancock College
BaumRA@aol.com
Corresponded by Richard A. Baum
319 West Valerio Street, Unit 4
Santa Barbara, California 93101
United States of America (USA)
The Helicoid as a Minimal Production Surface
and a Tension Minimizing Exchange and Production
General Equilibrium Model as a Soliton
JEL Classification:
Mathematical Economics and Welfare Economics
Keywords:
Minimal Surface, Helicoid, Production Surface, Satiation Point, General Equilibrium,
Product Differentiation, Factor Differentiation, Soliton of Fixed Points,
Perfectly Competitive Continuous Model of Monopolistic Competition,

2. 2
The Helicoid as a Minimal Production Surface and a Tension Minimizing
Exchange and Production General Equilibrium Model as a Soliton
Abstract
This paper develops a new model of monopolist competition. Ours will be a perfectly competitive
model with a continuum of perfectly differentiated products and a continuum of perfectly differentiate
Capital K and Labor L. In this model of perfect competition of a continuum of perfectly differential
products and a continuum of perfectly differentiated factors or resources, general equilibrium is a soliton,
not a single fixed point as presently conceived in the literature. Recall a soliton is a continuum of fixed
points of a flow. Further, in our model, such a flow of a soliton is found to be a tension minimizing
minimal surface. That is, the equilibrium core surface in our model is a minimal surface of equilibrium
tangencies between the production surface and an uncountable infinity of families of concentric spheres
modeling preference with the satiation of each family of spheres being the center of that sphere.
Section 1: Introduction
Donato [2] presents the first discussion in the literature of non-planar minimal surfaces in
economic theory of which the author is aware. This paper extends that discussion with regard to the
helicoid as it relates to production, and of families of spheres each with satiation point representing
preferences. In our model, there is a continuum of perfectly differentiated products and a continuum of
perfectly differentiated capital and labor. The main contribution of this paper to economic theory, and
where standard theory gets it wrong, is that in continuum economy general equilibrium is not a single

3. 3
fixed point, rather general equilibrium in a continuum exchange and production economy is a soliton [1]
when all products and factors are perfectly differentiated. Recall a soliton is a continuum of fixed points,
with first observation being height of a standing wave moving in a canal. A major contribution is rather
than the standard discrete model of monopolistic competition, where marginal cost equals marginal
revenue and is less than price, our model predicts given an isothermal parameterization of the production
surface, whereas price is positive, varying differentiated capital and labor pari passu, the marginal
revenue to each factor in equilibrium in a smooth context must both be zero. Ours will be a perfectly
competitive model with perfectly differentiated products and factors if we can show the marginal cost to
capital and the marginal cost to labor too are both zero given an isothermal specification of production,
with prices being actuarially fair or yielding only a normal rate of return. Further, in an Edgeworth Box
framework, our smooth core surface displays zero tension in general equilibrium among consumptive and
productive market participants. Given our model predicts this about an isothermal production function,
this work is left for a later paper. Rather, we develop solitons and this paper makes precise what we mean
by zero physical tension in an economic context.
Section 2: A Description of the Model and of Solitons
In 2019, the author rediscovered as applicable to economics as a production surface in economics
Meusnier’s 1776 discovery of the helicoid. Taking the unit normal to such a surface at every point, a
continuum of spheres each with different center or satiation point and each tangent to various points of the
production surface when the radius is unity of a member of each family of spheres may be used to model
preference. Hence a true new neoclassical school of economics is developed which models a smooth
continuum economy with a continuum of perfectly differentiated products and factors as being in
equilibrium as tension minimizing exchange and production of the two aforementioned geometric

4. 4
structures. That is the helicoid is used to model production and an uncountable infinity of families of
spheres is used to model preference, each family of which is a family of concentric spheres, with a
different center or satiation point for each family, however each member of the family having unit radius
when tangent to the production surface. The feature zero vector divergence of the unit radius at point of
tangency with the helicoid is what mathematically yields zero tension in the production surface and so too
yields a minimal surface to model production. The feature the production function may be modeled as a
level set is what yields the feature of a set of fixed points for the flow of the production surface, or more
simply, a set of equilibrium fixed points of a flow is called a soliton. The goal of this model is to yield a
tension minimizing solution to the problem of maximizing satisfaction given what society is capable of
producing.
Section 3: A Description of Zero Economic Tension and Harmonicity
What do we mean by zero economic tension on a smooth core surface? Two explanations are
given. The first uses imagery to make it accessible to the general reader. The second is mathematical,
which makes precise what we mean by zero economic tension when the core surface arises out of
productive and exchange activities given perfectly differentiated products and factors. Though the
foundations for a non-uniform density core have been laid in mathematics, to ease the presentation, we
assume the core surface displays uniform density. Lastly, we state our understanding of what it means for
the economy to be harmonic or in harmony or in a state of harmonicity.
Using imagery, consider yourself and your fortune, lot, or fate, as a point on a surface. In finance,
this point may be thought of as a position in the market. In economics, zero tension is the idea there is
neither an unwelcome crowding of, nor withdrawal from you, your fortune, or lot by your neighbors,
either in sentiment, sympathy or unwelcome attempt to exchange with you or your position. Recall in
standard Euclidean geometry, between any two points there is an infinity of points. Further, there is no

5. 5
point bumping right up against any point. Recall 0.5 0.4999 . They are the same number. So all
we are saying with this explanation using imagery is no point on the equilibrium core surface is an
accumulation point.
Now let us tackle the idea of harmony in equilibrium. Three conditions come to mind for our core
to be harmonic or the economy in a state of harmonicity. First, prices are fair. Given an actuarial study of
our economy, prices are actuarially fair. Second, as defined above, there is zero tension in our
equilibrium core surface. Lastly, and perhaps problematically, there is no envy. However, envy is people
trying to “get a piece of your action,” or simply trying to be you. Note simply wishing you were the
person envied has no empirical implications without some action taken based on this envy. Envy as
others trying to be you is ruled out by the following. First, given the above, no one is trying to crowd you
with their being or wants. Given a position on the equilibrium core surface, everybody is happy with their
position as there is no movement of any being away from their core position in equilibrium. Given
exchange and production, there may have been movement toward that core position, however, once
attained, zero tension implies no movement away from it.
Mathematically, zero tension is precisely the idea the vector divergence of the unit gradient to the
surface is identically zero. However, when the vector divergence of the unit normal to the surface
everywhere vanishes, we have a minimal surface. Typically when bounded by a loop of wire, soap films
in equilibrium minimize surface area among all surfaces possibly bound by the same loop of wire.
Mathematically, minimal surfaces are minimizers of the area functional or energy functional. This
is important when cost of doing business is proportional to the surface are of the production surface.
Section 4: A Smooth Tension Minimizing Exchange and Production General Equilibrium

6. 6
In this paper, we develop the author’s discovery of such a production surface as applicable to
economics of Meusnier in 1776 contemporaneous with Adam Smith’s 1776 publication of the Wealth of
Nations.
Recall in the classical case, we have Capital (K) and Labor (L) as variable factors of production.
Together, in this note, they produce the differentiated output ( , )z f K L which is a scalar valued
function of two variables. Recall in 1776 Meusnier discovered ([3], p. 16) an object named the helicoid
may be modeled in three real variables, ( , , )x y z as given by the function:
tan .
x
z
y

Alternatively, another helicoid is tan or ( , ) tanz x y z f x y x y   ([3], p. 26). The author uses
this latter specification where ,x K y L  and a continuum of perfectly differentiated outputs z is:
( , ) tanz f K L K L 
where capital 0K  and leisure or labor L is periodic (think of a pi clock) so .
2 2
L
 
  
In the following, consider only 0
2
L

  which is periodic labor. Though not considered in the
following, 0
2
L

   is called leisure. Then our production surface given as a graph of a function is the
map X where:
( , , ) ( , , ) ( , , ( , )) ( , , tan )X x y z K L z K L f K L K L K L    .
Again, consider the partial helicoid given functionally as tanz K L where 0K  and
0
2
L

  so the differentiated product z is never zero and divide both sides by z to give the level set:
tan
1 .
K L
z

Apparently points ( , , )K L z that satisfy this flow of the surface are fixed points of the level set or flow of

7. 7
of the production surface which implies a soliton. Further, when every point on this production surface is
just tangent to some sphere whose radius is one when tangent to the production surface that is general
equilibrium in our continuum production and exchange model. Apparently, our families of spheres
representing preference with satiation point the center of each member of a family of spheres has the
property that when the unsigned unit normal at every point of the production surface is the satiation point,
that is, the unsigned unit normal at every point on our partial helicoid is the unit radius of some family
member of spheres each with different center or satiation point however with radius one when tangent to
our partial helicoid minimal production surface, our production surface will everywhere have a tangency
with some unit sphere member of an uncountable infinity of families of spheres. Each family member is a
family of concentric spheres with center the satiation point of that family member. Notice we have not
been too particular about the sign of the unit normal to surface whose magnitude is the radius of the
sphere when tangent to the production surface. We will now be precise. Given zero vector divergence of
the unit normal to our production surface everywhere on our production surface, the production surface is
a minimal surface.
So too, as in general equilibrium, our soliton is fixed points of the flow, which contradicts present
thinking that general equilibrium is a single fixed point. As there is neither crowding nor withdrawal
from your position, lot or fortune, we may claim to have minimized economic tension.

8. 8
References
[1] Baum, Richard A., Does it make Sense to call this a Soliton?, Mathematics Stack Exchange,
Posed by Richard Anthony Baum, August 30, 2019.
[2] Donato, Jerry, Minimal Surfaces in Economic Theory, in A. Prastaro and T. M. Rassias (Eds.),
Geometry in Partial Differential Equations, pp. 68-90, World Scientific, 1994.
[3] Morgan, Frank, Riemannian Geometry: A Beginner’s Guide, Second Edition, A. K. Peters, 1998.