Positivity 6: 201–204, 2002.
© 2002 Kluwer Academic Publishers. Printed in the Netherlands.
Introduction to the special issue on mathematical
C. D. ALIPRANTIS
, B. CORNET
and R. TOURKY
Department of Economics, Purdue University, West Lafayette, IN 47907–1310, USA (E-mail:
CERMSEM, Maison des Sciences Economiques, Université Paris I, 106–112 Boulevard de
l’Hopital, 75645 Paris Cedex 13, France (E-mail: firstname.lastname@example.org)
Department of Economics, University of Melbourne, Melbourne, VIC 3010, Australia (E-mail:
This special issue of Positivity is devoted to mathematical economics. Its ob-
jective is to expose the reader to topics in economic theory that use advanced
techniques from analysis and topics in mathematics that arise from problems in
economic theory. Special emphasis is given to techniques that are associated with
order structures and positivity. The issue contains eight papers.
1. Economic Equilibrium: Optimality and Price Decentralization, by C. D.
Aliprantis, B. Cornet, and R. Tourky
This is a survey article that summarizes topics on general economic equilibrium
that models the interaction of supply and demand and which postulates a price
equilibrium where supply is equal to demand in every market.
The emphasis of the article is two-fold. First, the theory of general economic
equilibrium with inﬁnitely many commodities, where typically the analysis relies
on vector lattice theoretic arguments. Second, general equilibrium theory with a
non-convex production sector, a ﬁeld which has beneﬁted from the development of
new mathematical techniques known as “Nonsmooth Analysis.”
2. Euler Characteristic and Fixed Point Theorems, by B. Cornet
In a smooth setting, the well-known Poincaré-Hopf theorem, or classical degree
theory, implies that every continuous mapping f : M → R
which is “tangent”
to M admits an equilibrium (zero), i.e., x
∈ M and f(x
) = 0, whenever M is
smooth and its Euler characteristic is nonzero; this applies, in particular, to convex
sets via a simple approximation argument, recalling that the Euler characteristic of
a convex set is one. Such results are particularly important in economic equilibrium
analysis, ﬁrst for the existence problem, and second to have more information on
the equilibrium set, for example by “counting” the number of equilibria modulo 2.