A description of tension minimizing economies that naturally form minimal surfaces.

1. Tension Minimizing Economies are Minimal Surface Economies August 2, 2017
Copyright © 2018, Richard Anthony Baum
Lecturer, Department of Economics, Retired
Allan Hancock College
One Hancock Drive
Lompoc, CA 93436
BaumRA@aol.com
Keywoods: Minimal Surfaces in Economics, Tension Minimizing Economies
Isothermal Minimal Surface Economies and Core Equivalence
ABSTRACT
Using differential geometry, a description of an economy that exhausts gains from trade
while preserving a vector divergence-free evolution of the Euclidean unit normal price vector is
given for the case of a public classical and a public signaling economy. The purpose is to to
minimize tension between resource owners and consumptive market participants. Though
limited to Euclidean 3-space, it easily generalizes to higher dimensions by a divergence-free
evolution of the Euclidean unit price vector. Assuming a point is not umbilic, a Gram-Schmidt
orthonormalization process may be numerically implemented once a principal direction has been
determined which is an eigenvector whose eigenvalue achieves a maximum or minimum at that
point. Solving analytically or by iteration, or by algorithm in a neighborhood of a point with
boundary, when the system always maintains zero tension, the surface representing it is a

2. minimal surface which minimizes area costly information must span. Exchange and production
is at every point a point of harmonicity in a minimal surface economy.
INTRODUCTION
Armen A. Alchian (see [1],[2],[7]) was the author’s graduate microeconomics 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 first discussion in the literature of minimal surfaces in economics of which the author
is aware was in 1994 in an article by Donato [A]. The present paper is an extension of work
conceived February 20, 2015 by the author. On September 1, 2015, Aydin and Sepet [B]
submitted their paper on minimal production surfaces using Galilean Geometry. On March 1,
2016, Aydin and Ergut [C] posted to Math arXiv a discussion of production functions in
economics with a brief reference to when mean curvature 0H  , which would imply a minimal
surface.
Minimal surfaces have a long and rich history in mathematics, however are only
beginning to appear in economics. The author began his study of them in 1987 while a student

3. of differential geometer John Douglas Moore at the University of California Santa Barbara,
realizing the tangent wealth hyperplane to an indifference surface is a trivial minimal surface. It
was later found a pencil of hyperplanes is a minimal submanifold (a 4-dimensional extension
equivalent to fully opening a book along its spine where each page is a level set) or space itself
less the origin when every hyperplane passes at any angle through the origin.
Each hyperplane corresponds to a potential, the origin being excluded because it
corresponds to a point of infinite potential. The potential of each punctured hyperplane may be
thought of as its energy or the real wealth corresponding to that wealth hyperplane. Recall a
wealth hyperplane is analogous to a budget constraint in elementary microeconomic theory.
The topic of this tribute in honor of Armen A. Alchian is minimal surface economies and
core equivalence. Planar economies have been characterized in the literature as minimal
surfaces. The author believes 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 1 2 3( , , )x x x , 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

4. 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 bundle 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 bundle 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.

5. 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 bundle 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.

6. Let ( , )u v be orthogonal coordinate axes representing the two public factors of
production. Then, for saddle surfaces, orthogonal cross-sections with respect to ( , )u v are
simultaneously partially convex and concave. A three good public economy may be
parameterized as 1 2 3( , ) ( ( , ), ( , ), ( , ))u v x u v x u v x u vx . Recall from the third paragraph, the
surface ( , )u vx 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 ( , )u v and a target space of ( , ).u vx
( , )u vx may be embedded in 3
. However, while the source space is not necessarily Euclidean,
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 3
its surface may be immersed as the two-sphere 2
.
The discussion is easy. The following requires some mathematical rigor.
ANALYTICS

7. Consumption space 1 2 3( , , )x x x denominated in the unit of account, say dollars, euros or
yen, where one is chosen as numéraire (the numéraire is the unit of account in terms of which all
other unit of accounts are relative or measured in terms of), is parameterized by ( , )u v .
Consider the surface 1 2 3( , ) ( ( , ), ( , ), ( , )).u v x u v x u v x u v x x Assume throughout ( , )u vx is a
regular parameterized surface.
DEFINITION 1: ( , )u vx x is isothermal if , ,u u v vx x x x and , 0u v x x , where
, denotes the standard inner product.
DEFINITION 2: ( , )u vx x is a minimal surface if its mean curvature is zero.
DEFINITION 3: ( , ) ( 1,2,3)ix u v i  is harmonic if 0ix  where  is the Laplace
operator defined by
2 2
2 2
.
u v
 
  
 
THEOREM 1: Assume x is isothermal. Then x is a minimal surface if and only if
 x 0 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 1 2 3( , ) ( ( , ), ( , ), ( , ))u v x u v x u v x u vx be consumption space. Let x* be an

8. 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,
* * (0,0,0)uv vu  x x 0 and * * (0,0,0) .uu vv  x x 0 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 * and *uu vvx x 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. * and *u vx x
vectors of marginal value factor returns at optimas. Think: does theory say one ought to
exploit gains from trade until the length of the change in marginal value factor returns
come into equality, that is * *uu vv  x x , and is * *uu vv x x 0 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 *x are points of
harmonicity given they are points on an isothermal production surface, which is to say
the optimal production surface given harmonicity is an isothermal minimal surface.
DIGRESSION BEFORE THE COUP DE GRACE:

9. Economic theory says to produce until the marginal revenue products of u and v come
into equality. Formally, we produce until * *u vx x . 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 * *u vx x . 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 *, * *, * ,u u v v   x x x x some positive constant, it
follows:
*, * *, * 0 so * * *
*, * *, * 0 so * * * .
u u uu u uu u uu
v v vv v vv v vv
u
v

    


    

x x x x x x x 0
x x x x x x x 0
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, assuming we have identified a
principle direction, trivially yields the other principle direction such that
*, * *, * 1 and *, * 0u u v v u v  x x x x x x . We assume points of production
are regular so the first preceding equality implies the inner products are non-zero, while
the second equality implies * *u vx x . 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 3
, namely
 
* *
*, *, * where * .
* *
u v
u v
u v



x x
x x n n
x x

10. END OF DIGRESSION
Given this, ( , )u vx admits the parameterization ( , ) *u v u v   x x 2
(( , ) )u v R
where  and  are orthonormal vectors on ( , ).u vx That is, 1   and
, 0.   It follows * , * ,u u v v    x x x x and * *uu vv uu vv   x x x x 0 if
*x is a point of harmonicity. Hence,
(i) , ,u u v vx x x x and
(ii) , 0u v x x
so that consumption space is isothermal. Further,
(iii) ( , ) uu vvu v   x x x 0
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 *x . 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 ( , , ( , )) 1,2,3iu v x u v i  .
Recent discussion (see Donato [A]) has focused on the Cobb-Douglas production
function and has characterized inputs ( . )u v as being perfectly non-substitutable, and thereby
having infinite elasticities of substitution. The classical case of this in economics is the Leontief

11. production function. In the plane, it forms a series of production functions forming right angles
parallel to the axes, the corner each of which just touches an infinite foliation of budget
constants, each corresponding to different expenditure but each having constant slope.
Apparently, for an infinitesimal change in u, the corresponding change in v is infinite, and for an
infinitesimal change in v, the change in u, is infinite, while maintaining the level set implied by
each of the right-angle Leontief production functions, each of which is projected onto the uv
plane as an isoquant.
Extending Colding and Minicozzi’s 2011 treatment where every embedded minimal
surface may be broken down into parts of plane or parts of a double sided spiral staircase (see
[D], pp. 289-94), it is seen the three dimensional analogue to such a production function is the
ruled surface called the helicoid. Using a Frenet Apparatus of a moving orthonormal frame, we
see the Leontief productions functions rotating upward along each step of the spiral staircase,
which in the limit as step size approaches zero is a very nice smooth approximation of what had
been discrete steps into a smooth ascending helicoid, which is a classical minimal surface.
To see this, perform a rigid motion of the double spiral staircase so as to achieve an
orientation consistent with the above description. Let the ambient space be xyz space with the
usual choice of axes according to the right hand rule. For each choice of u, a choice is made of a
v ruling parallel to the xy plane in xyz space. As u changes, this ruling rotates upward or
downward, this new ruling still parallel to the xy plane with axis of rotation being the z axis in
xyz space. The amount of movement of this spiraling surface along the z axis is given by the
choice of v. A ( , )u v parameterization of this surface embedded in xyz space is

12. ( , ) ( cos , sin , ) ( , , ) where and .u v v u v u v x y z u v          x x
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 exogenously 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 *x (each of which minimizes tension among market
participants, that is a point of harmonicity) and exogenous directly observable types
( , )  is a submanifold of 3
given by ( , ) *u v u v   x x . The problem is to
prove ( , )u vx is a minimal submanifold. Given {( , )}  such that 1   and
, 0   , then as before we have *u u  x x , *v v  x x ,
* *uu vv uu vv   x x x x 0 and ( , ) uu vvu v   x x x 0 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
* and *u v  x x . Then for each orthonormal pair ( , )  there is a minimal
surface and a minimal submanifold results from {( , )}  . In summary, at any points
1 2 3( , ) and ( , , )x x x  , there are 3 3 6  signals (recall each and  are 3-tupples)

13. corresponding to each of the 3 2 6  ways of permuting
1 2 3( , ) ( ( , ), ( , ), ( , ))u v x u v x u v x u vx at any point ( , )u vx in our target sculpture of
consumption space residing in 3
. This corresponds with the 1980’s notion there ought
to be one signal for each resource-consumption good combination.
Each tangent space of each of the above two types of economies is given by Span
( , )u vx x . Given regular parameterized surfaces, the unit normal price vector at any point on
either surface is ( , ) ( , ).u v
u v
u v
 
 
 
 
  
 
x x
n n
x x
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 { ( , )}u vn for some ( , )  .
DISCUSSION
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:
t

 

x
x
where time t > 0 . This system of equations is realized by a squeezing of a reaction-diffusion
system of partial differential equations:
( )
t

  

x
x f x

14. 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 (0,0,0)0 . 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:
 0,0,0
t

   

x
x 0 .
The only difference is the free energy which caused the shock to the initial minimal surface
imparts free energy to the system to do useful work, so the energy of our surface (where in the
case of a two-dimensional minimal surface, the energy is the area of the surface) degrades in the
process of adjusting into the unavailability of energy to do useful work, this type of degradation
of energy to do useful work leading to the ever increasing entropy postulated by
thermodynamics, statistical mechanics and fluid dynamics, to name a few other sciences. Thiel

15. [E] in his work on continuity and information theory in economics speaks of informational
entropy, following Shannon [F].
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 ( , )u v x 0 where it is understood an equilibria is such that
t



x
0 in
equilibria.

16. 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 1 2 3x x x    be real wealth at 1 2 3( , , )x x x . 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 ( , )u v 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

17. 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 ( , )u v 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 u v
u v



x x
n
x x
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
2 2
div (1 ) 2( ) (1 ) (0,0,0)v uu u v uv u vv       n n n n n n n n 0
(see Ito [15], p. 1031) where it is understood 1 2 3:( , ) ( ( , ), ( , ), ( , ))u v x u v x u v x u vn and each
( , ) 1,2,3ix u v i  is a graph map with coordinates ( , , )iu v x . Let div n be defined for this

18. graph map 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 0  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

19. 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

20. [A] DONATO, J., Minimal Surfaces in Economic Theory, in A. Prastaro and T. M. Rassias
(Eds.), Geometry in Partial Differential Equations, pp. 68-90, World Scientific,
1994.
[B] AYDIN, M. E. & M. ERGUT, Isotropic Geometry of Graph Surfaces Associated with
Product Production Functions in Economics, arXiv: 1603.00222v1,
[math.DG], 1 Mar 2016 (online mathematics repository).
[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 of
General 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 Nations
BY 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, Second
Edition, 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, American
Journal 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

21. [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 Related
Screening Processes, Harvard, Cambridge, MA 1974