A valid production model which forms a minimal surface in economics. Though UCLA, Harvard and the University of California at Santa Barbara have been notified, it will likely be relegated to the dustbins of history without some proper incentive structure.

1. Richard Anthony Baum February 16, 2019
Department of Economics, Semi-Retired
Allan Hancock College
BaumRA@aol.com
Helicoid as a Minimal Surface Production Surface
In 2019, Baum rediscovered Meusnir’s 1776 discovery of the helicoid as applicable to economics
as a production surface in economics. It is conjectured its conjugate surface, in the first octant, namely
the catenoid, may be used to model preference, and hence a true new neo-classical school of economics is
developed which seeks to model a smooth continuum economy as being in equilibrium as tension
minimizing exchange and production of the two aforementioned minimal surfaces, namely the helicoid
for production and the catenoid for exchange to yield a tension minimizing solution to the problem of
maximizing satisfaction given what society is capable of producing.
In this short note, we develop Baum’s discovery of Meusnir in 1776 contemporaneous with Adam
Smith’s publication of the Wealth of Nations of such a production surface. After research, an anticipated
following note will model exchange.
Recall in the classical case, we have Capital (K) and Labor (L) as variable factors of production.
Together, in this note, they produce a single output ( , )z f K L which is a scalar valued function of two
variables. Specifically, in the case of Meusnir, the production map or surface given as the graph of real
valued variables,
1
( , , ) ( , , ( , )) ( , ,tan ).
K
X K L z K L f K L K L
L
  
    
 
Perhaps this is too difficult a place to start. Recall in 1776 Meusnir discovered an object named
the helicoid may be modeled in three variables, ( , , )x y z as given by the function

2. tan .
x
z
y

Our model follows by letting ,x K y L  and solving for z and expressing the
production surface 1
( , , ) ( , , ) ( , , ( , )) ( , ,tan )
K
X x y z K L z K L f K L K L
L
  
     
 
as the map given by
the graph of a function.
Recall from Paul A. Samuelson and William D. Nordhaus 19th
edition of Economics, copyright
2010 by McGraw-Hill Irwin publishers that a productive equilibrium is characterized by distribution such
that
1
.K LMP MP
r w MR
 
Here, , , , , andK LMP MP r w MR are the marginal product of capital, the marginal product of labor,
the rental price of capital, the wage rate, and the marginal revenue generated in selling the product. Note
another way of expressing this relation is
1.K LMP MR MP MR
r w
 
In the above equation, note we are dealing with scalars, not vectors. Taking the inner product of
each above part of the expression (recall the inner product is simply what economists call the dot
product), we have
, 1
, 1
K K
L L
MP MR MP MR
r r
MP MR MP MR
w w



3. For those unfamiliar with the inner product notation above, recall it is the same as the dot product
where ,F F F F  where F is allowed to be either a scalar or a vector.
Given our map or surface ( , , ( , ))X K L f K L , we would like our surface to be isothermal,
namely
, , and , 0K K L L K LX X X X X X 
Now
(1,0, ) (1,0, )
(0,1, ) (0,1, )
K K K
L L L
X f MP
X f MP
 
 
and , (1,0, ),(0,1, ) 0K L K L K LX X f f f f  
if and only if for all 0K Lf f  the above implies K Lf f , that is, Kf and Lf are orthogonal
functions.
How do we prove this?
Recall
K K
L L
r
MP f
MR
w
MP f
MR
 
 
so
, , 0K L
r w
f f
MR MR
 
for the case of orthogonality of the two component functions.
What do we mean by MR, that is marginal revenue. Marginal revenue is defined as the derivative
of total revenue with respect to quantity. Does this definition need refining in the case of the above
middle inner product. The author answers affirmatively. Now total revenue, TR, is universally defined as
( , , ) ( , , ( , )) ( , )TR pq pz pf p K L z z p K L f K L f K L     where p is the price of the product and
( , )q z f K L  is quantity of the single output.

4. Accepting this, then total revenue is a function of several variables, namely price, capital and
labor as output is explicitly a function of capital and labor. Logically, there is marginal revenue given as
a function of K, which we denote KMR , and marginal revenue given as a function of L, which we denote
LMR .
Let us explicitly define each term.
( )
( )
K K K
L L L
TR pf
MR p f pf
K K
TR pf
MR p f pf
L L
 
   
 
 
   
 
which follows from the product rule. We would like to show
, 0.
K K L L
r w
p f pf p f pf

 
To do so, we appeal to reason. What is the inside of this inner product? It is the additional distribution
to factors capital and labor that can be squeezed by altering both capital and labor at the same time.
Recall the above condition holds in general equilibrium. In general equilibrium, though not quite the
production exhaustion law which is known of Euler’s condition, it is the factor exhaustion law that no
additional distribution can be squeezed from factors pari passue (changing both factors) and not simply
the usual condition of ceteris paribus (changing only one factor). Q.E.D.
The above represents a new condition for general equilibrium in a neoclassical economy. For
evidence the above surface is a minimal surface which minimizes tension in equilibrium, refer to Frank
Morgan, Riemannian Geometry: A Beginner’s Guide, Second Edition, copyright 1998 by Frank Morgan,
published by A. K. Peters.