Richard A. BaumSanta Barbara, CA 93101February 20, 2015BaumRA@aol.com

Isothermal Minimal Surface Economies

and Core Equivalence

Armen A. Alchian (see [1],[2],[7]) was my graduate economics principles instructor at

UCLA in 1974. Armen passed away February 19, 2013, of natural causes at age 98. May his

spirit rest in happy peace. Professor Alchian studied mathematics and economics at Stanford

University where he was granted a Doctor of Philosophy degree in 1942.

The topic of this tribute is minimal surface economies and core equilivance. Planar

economies have been characterized in the literature as minimal surfaces. I believe part of

Professor Alchian’s class discussion was on cores of planar economies. The economic relevance

of such a discussion is to minimize tension among market participants after having exploited all

gains from trade. The economic relevance of this paper is to minimize tension among market

participants in consumption space if we may first allow mutually offsetting gains in production

space. Such is the theory of perfect elasticity when a metric in (u,v) space, where say u and v are

labor and capital, is preserved in some target, say consumption goods , by a process

of stretching or shrinking our source (u,v) of resources to fit our target consumption sculpture so

as to achieve a perfectly elastic equilibrium in the rubber cap stretched over our consumption

sculpture. Elastic equilibrium implies zero tension at every point in our rubber cap (see Eells

and Lemaire [11] page 1).

This paper is also a geometric treatment of parts of Balasko’s equilibrium manifold [4]

and Spence’s notion of signaling [19].

Except for planar surfaces, a lot of minimal surfaces form some sort of saddle. This is

consistent with having zero or negative Gaussian curvature. Minimal surfaces have zero mean

curvature. The amount of bending of the surface by some sort of shape operator averages out to

zero in the small and the large. Tangent to each point of a minimal surface is a plane.

Obviously, planes have zero net bending, or zero mean curvature. The set of all tangent planes

to a minimal surface is the tangent space of the minimal surface.

As the unit normal (see Appendix) to some sort of plane in economics seems important to

establish prices at which agents may move along the tangent plane and obtain their desired

bundle of goods, the tangent space becomes important when there are a multitude of equilibria

each existing at different price. This seems to violate the rule of one price, to which a placebo

has been offered in the form of a multitude of signaling equilibria, each position being offered to

some type of agent at some price, not all necessarily the same. An older way of violating the

rule of one price is with price discrimination, typically practiced by many doctors including

economists.

What is new in this paper is a proof classical competitive regular public economies, be

they one or many price economies with public factors of production, form minimal surfaces. A

corollary establishes that modern competitive signaling regular public economies with public

signals may also be minimal surfaces. Regular surfaces are smooth (no corners) whose normal at

every point is non-zero, consistent with the unit price vector not being the zero vector. Though

the difference between price and opportunity cost in a perfectly competitive private economy is

zero, economies with prices being the zero vector and the opportunity cost vector being the zero

vector are plain and not very interesting.

Also new is a type of core equivalence between parts of classical perfectly competitive

regular public economies and parts of modern perfectly competitive signaling regular public

economies. This is due to a correspondence between the tangent planes of the parts. Though

positions do not necessarily coincide, what is typically meant by a core is different positions

trace out a path, an area, a volume, or in this case a correspondence between an evolution of unit

normals to tangent planes of parts of each of the above two regular economies which are in fact

the unit price vectors in each economy. When the unit price vectors coincide, we have a form of

core equivalence for the two economies. This should not be taken to mean the tangent space of

both economies coincide.

What is not new is economies with a continua of traders and a continua of goods. Also

not new is dense economies. Since the economies under discussion are for the most part saddle

shaped, gross complementarity does not arise among isoquants with two public factors of

production.

Let be orthogonal coordinate axes representing the two public factors of

production. Then, for saddle surfaces, orthogonal cross-sections with respect to are

simultaneously partially convex and concave. A three good public economy may be

parameterized as . Recall from the third paragraph, the

surface may be planar or some type of saddle. I believe gross complementarity typically

arises with planar surfaces. With saddle surfaces it is not a problem as the orthogonal cross

sections of the isoquants imply some degree of simultaneous substitutability and

complementarity.

Recall we have a source space of public factors and a target space of

may be embedded in However, while the source space is not necessarily euclidian, a

metric on our target space may be pulled back isometrically to yield an isometric minimal

(harmonic) immersion of our source into our target 2-surface. Indeed, much as the 3D ball may

be embedded in its surface may be immersed as the two-sphere .

The discussion is easy. The following requires some mathematical rigor.

Consumption space denominated in the unit of account is parameterized by

. Consider the surface Assume throughout

is a regular parameterized surface.

DEFINITION 1: is isothermal if and , where

denotes the standard inner product.

DEFINITION 2: is a minimal surface if its mean curvature is zero.

DEFINITION 3: is harmonic if where is the Laplace

operator defined by

THEOREM 1: Assume x is isothermal. Then x is a minimal surface if and only if

where 0 = (0,0,0).

PROOF: See Carmo [9], pp. 201-202.

THEOREM 2 (EXISTENCE): Consumption space in a classical, perfectly competitive full

information non-signaling, economy is an isothermal minimal surface.

PROOF: Let be consumption space. Let x* be an

optimal production point which for convenience we suppose minimizes tension among

market participants and so is a point of harmonicity. Forming matrices of second partials,

it follows from the theory of elasticity that minimizing tension implies traction (off

diagonal terms) is zero and the trace of the matrices of second partials is zero. That is,

and The implication is the rate of

change of the marginal value products of the two factors have equal magnitude but

opposite sign. Geometrically, this implies are mutually offsetting vectors

or forces in the market. We would kind of expect that if an optimal point of production

were to minimize tension in the market among market participants. A simple way of

seeing this is to draw an Edgeworth box for production and observing when all gains

from trade have been exhausted, normals to tangent isoquants in the Edgeworth box point

in opposite directions. What remains to show is they have equal length.

are kind of like marginal value factor returns at optimas. Think: does theory say one

ought to exploit gains from trade until the length of the marginal value factor returns

come into equality, that is , and is kind of like some

type of second order optimizing condition for critical points of optimal levels of

production: it certainly is if one is trying to minimize tension between resource owners

and consumptive market participants. This is precisely the case when are points of

harmonicity.

DIGRESSION BEFORE THE COUP DE GRACE:

Economic theory says to produce until the marginal revenue products of u and v come

into equality. Formally, we produce until . Scarcity of resources tends to

imply this is non-zero while our assumption of regularity assures this is not zero. Also,

we would like to show . The standard inner product is self adjoint and

normal (see Axler [3], pp. 99-100 and pp. 128-133) with respect some operators, for

example differentials and partial differentials (see Debnath and Mikusinski [10], pp. 247-

248). For example, from some positive constant, it

follows:

Further, Gray [14] shows any system of coordinates on a minimal surface can be made

isothermal. Assuming non-parallel vectors, another way of securing this result is with a

Gram-Schmidt orthonormalization process which yields

. We assume points of production

are regular so the first preceding equality implies the inner products are non-zero, while

the second equality implies . In summary, x* having been made isothermal

using a Gram-Schmidt orthonormalization process on the two vectors of first partials

immediately above produces an orthonormal vector basis for , namely

END OF DIGRESSION

Given this, admits the parameterization

where and are orthonormal vectors on That is, and

It follows and if

is a point of harmonicity. Hence,

(i) and

(ii)

so that consumption space is isothermal. Further,

(iii)

so that consumption space is a minimal surface.

In this case, consumption space is simply a submanifold of wealth hyperplanes, one

wealth hyperplane passing through each of the various . If, in addition, points of optimal

production which minimize tension form a smooth regular surface, the surface x* above is a

minimal surface not necessarily planar, but rather saddle shaped with negative Gaussian

curvature. Classical candidates for a minimal production surface are the helicoid and the

catenoid, or perhaps even three Scherk’s first surfaces .

COROLLARY (EXISTENCE): Consumption space in a perfectly competitive signaling economy

in stable equilibrium is a minimal surface given a set of orthonormal signals.

PROOF: Signaling implies equilibrium depends on signaling types where vary,

and may be exogenously given or endogenously determined by the system. Suppose

signaling types are endogenously determined by the system and have the feature they are

observable while productivity itself may or may not be directly observable, especially

when there are joint factors of production. Be that as it may, a parameterization of

consumption space in a perfectly competitive signaling economy in stable equilibrium for

various optimal production points (each of which minimizes tension among market

participants, that is a point of harmonicity) and endogenous directly observable types

is a submanifold of given by . The problem is to

prove is a minimal submanifold. Given such that and

, then as before we have , ,

and where x* are points of

harmonicity. However, we now assume that productivity is a constant scalar multiple

of signals. That is, we assume observable signals can be realized such that

. Then for each orthonormal pair there is a minimal

surface and a minimal submanifold results from . In summary, at any points

, there are signals (recall each are 3-tupples)

corresponding to each of the ways of permuting

at any point in our target sculpture of

consumption space residing in . This corresponds with the 1980’s notion there ought

to be one signal for each resource-consumption good combination.

The tangent space of each of the above two types of economies is given by Span .

Given regular parameterized surfaces, the unit normal price vector at any point on either surface

is Hence, the core equivalence of the two economies

given the set of all common to both economies is the set of all unit price vectors traced

out by common , namely for some .

Seemingly, the analysis has been static so far, or perhaps autonomous (time independent).

We may explore a global dynamical setting (see Smale [18] and Eells [12], [13]) by allowing

diffusion of a perturbation to the economy according to the diffusion or heat partial differential

system of equations, to wit:

where time t > 0 . This system of equations is realized by a squeezing of a reaction-diffusion

system of partial differential equations:

where f(x), the reaction term, is characterized as excess demand. We assume opportunistic

behavior is led as though by an invisible hand to squeeze all super-profit or loss out of the system

so as to leave it in a state of equilibrium where excess demand is . Let t = 0

correspond to an initial equilibria. Then let there be an exogenous or endogenous shock at some

0 < t < 1 . Assume it be closer to 0 than 1, perhaps infinitesimally close to 0 . The minimal

surface representing the initial equilibria is no longer minimal due to the act of being deformed

from its original shape. Perhaps economically and physically, the surface representing

consumption space is sent into wild vibrations. Adam Smith in The Wealth of Nations observed

opportunistic behavior by individuals is led, as though by an invisible hand, to an outcome that

benefits all participants (see [18], p. 477). This opportunistic behavior continues until all the

gains from trade have been exhausted. By assuming certain regularity conditions, our slowly

diffusing consumption surface reacts so the free energy released by the shock is gradually

degraded by increasing entropy into unavailable energy, so our surface which initially reacted

now diffuses until it winds down into a new steady state equilibria at an arbitrary time 0 < s + t <

1 , s > 0 . The scaling of time is pretty arbitrary. The initial and new isothermal steady state

equilibria are minimal surfaces as:

.

One may speculate should progress proceed at unit speed and opportunistic behavior is

efficient, a path from initial to new equilibria is a geodesic. This is perhaps true when movement

along the path on the surface is constrained to move in the tangent space by action of the unit

normal to the surface at each point p. This unit normal may be thought of as the unit price vector

at which trades may be made at that point p. Trade along the surface according to movement on

a path constrained by the unit normal to the surface form geodesics which in general minimize

arc length from one point to another on the surface (see Pressley [17], p. 171).

Progress at unit speed and efficient trade minimizing arc length between two positions on

our surface with the additional regularity assumption our surface does not blow up or collapse in

the process, perhaps due to surgery or the intervention of central banks or regulators when

things get too catastrophic, may insure our surface, though initially anchored by its initial value

boundary at worst suffers a lift (or fall) with a little stretch until re-anchored by the law of the

land and exhaustion of gains from trade until it reaches its new position of steady state equilibria.

What started as a minimal surface gets mapped by the near catastrophe into a new, somewhat

different, minimal surface. In the steady state, the preceding proofs still hold as the steady state

is characterized by where it is understood an equilibria is such that in

equilibria.

Perhaps Professor Alchian would have understood. I am relatively certain Professor John

Forbes Nash, Jr. would have had no problem as an economist.

APPENDIX

Three normalizations of price frequently used in economics literature (See Balasko [5]

page 3) are:

(1) Choose a numéraire and set it equal to unity,

(2) The unit simplex method, and

(3) The unit euclidean norm method.

A most natural normalization that allows all currencies or goods to serve as numéraire is the unit

normal. Let be real wealth at . Note these variables are not

necessarily independent of one another.

So what are we to do? Give up? No, life is not that simple. For the sake of sanity, let us

simply assume variables exist independently of the others in the static (here and now)

setting. Recall we have chosen to call u and v labor and capital. We will not pick the argument

that labor is independent of capital and vice versa, however rather rely on the mathematical logic

of the previous sentence and settle ourselves with the assumption that the two variables u and v

are independent without necessarily arguing capital and labor are independent. We have not

logically contradicted ourselves, however made life a little simpler. To clear things up, the

variables u and v are independent while the names and concepts capital and labor or labor and

capital are not necessarily independent. Indeed a now deceased Nobel Laureate in Economics

was brave enough to talk of human capital (see Becker [6]). So let us content ourselves with

talking about the two independent variables without worrying too much whether they are

presently alive.

When x is parameterized by (u,v), the unit normal price vector to the surface given in

terms of the factors of production is simply the unit normal which maps the

surface nicely onto the unit sphere under the Gauss map.

Another way of saying our economy’s submanifold is minimal or in equilibrium is when

the unit normal price vector n is divergence free. Recall the Gauss map for regular surfaces

resides on the surface of the unit sphere, where some points of the unit sphere are omitted

because they correspond to points where there is not a unit normal to the surface. Say

(see Ito [15], p. 1031) where it is understood and each

is a graph map with coordinates . Let div n be defined as above.

The payoff comes when, as Nash (see [16] p. 5) has suggested, we may possibly have small

remembrances of the past and possibly small inklings as to the future, knowing the future is

inherently uncertain. That is, our analysis takes place on the time interval where is

a small number with mean zero.

VISIONS OF THE FUTURE

For those ready to tackle the problem, this may bring to mind our previous discussion of

mean curvature, where for a minimal surface the mean curvature or tension field identically

vanishes. That happens precisely when the divergence of the unit normal to our surface or what

is the same thing, the divergence of the unit price vector taken with respect to (u,v), is zero or

divergence free. In the case adjustment is instantaneous, the result is we have a family of

isothermal minimal surfaces when we constrain movement due to shocks to naturally minimize

tension in the price process so as to preserve (1) a divergence free unit normal price evolution,

(2) a zero mean curvature evolution of each family member surface representing an economy and

(3) a minimality of our model in the sense families of surfaces which arise from shocks exhaust

gains from trade due to shocks so as to preserve a divergence free Gauss map process of unit

normals to each family member surface or, what is the same, minimize tension among all market

participants be they resource owners or consumers in the course of exploiting gains from trade

due to shocks. In the case adjustment is instantaneous, we have an elliptic system as evidenced

by the spectrum of the Laplace operator. In the case adjustment takes time, our system is

parabolic and may be modeled by a system of reaction-diffusion equations eventually resulting

in zero excess demand (vanishing reaction terms) so as to achieve the heat or diffusion system in

the steady state and the Laplace system in the stationary or autonomous state when shocks cease

to be a disturbing factor.

REFERENCES

[1] A. A. ALCHIAN & W. R. ALLEN, University Economics: Elements of Inquiry, Third Edition, Wadsworth, Belmont, CA, 1972

[2] A. A. ALCHIAN & W. R. ALLEN, Exchange and Production: Competition, Coordination,and Control, Second Edition, Wadsworth, Belmont, CA, 1977

[3] S. AXLER, Linear Algebra Done Right, Second Edition, Springer, New York, NY, 1997

[4] Y. BALASKO, The Equilibrium Manifold: Postmodern Developments in the Theory ofGeneral Economic Equilibrium, MIT Press, Cambridge, MA, 2009

[5] Y. BALASKO, General Equilibrium Theory of Value, PUP, Princeton, NJ, 2011

[6] G. S. BECKER, Human Capital: A Theoretical and Empirical Analysis with Special Reference to Education, Third Edition, University of Chicago Press,

Chicago, IL, 1993

[7] D. K. BENJAMIN, ED, The Collected Works of Armen A. Alchian, Volumes 1 and 2, Liberty Fund, Indianapolis, IN, 2006

[8] E. CANNAN, ED., An Inquiry into the Nature and Causes of The Wealth of NationsBY ADAM SMITH, University of Chicago Press, Chicago, IL, 1976

[9] M. P. DO CARMO, Differential Geometry of Curves and Surfaces, Prentice-Hall,Englewood Cliffs, NJ, 1976

[10] L. DEBNATH & P. MIKUSINSKI, Introduction to Hilbert Spaces with Applications, SecondEdition, Academic Press, San Diego, CA, 1999

[11] J. EELLS & L. LEMAIRE, Two Reports on Harmonic Maps, World Scientific, Singapore, 1995

[12] J. EELLS & J. H. SAMPSON, Harmonic Mappings of Riemannian Manifolds, AmericanJournal of Mathematics, 86, 1964, pp. 109-160

[13] J. EELLS, A Setting for Global Analysis, Bulletin of the American Mathematical Society,72, 1966, pp. 751-807

[14] A. GRAY, Modern Differential Geometry of Curves and Surfaces with Mathematica, Second Edition, CRC Press, Boca Raton, FL, 1998

[15] K. ITO, Encyclopedic Dictionary of Mathematics, Second Edition, Volumes 1-4. MIT Press, Cambridge, MA 1987

[16] H. W. KUHN & S. NASAR, ED., The Essential John Nash, PUP, Princeton, NJ 2002

[17] A. PRESSLEY, Elementary Differential Geometry, Springer, London, 2002

[18] S. SMALE, The Mathematics of Time: Essays on Dynamical Systems, Economic Processes, and Related Topics, Springer-Verlag, New York, NY, 1980

[19] A. MICHAEL SPENCE, Market Signaling: Informational Transfer in Hiring and RelatedScreening Processes, Harvard, Cambridge, MA 1974