Positivity 6: 205–241, 2002.
© 2002 Kluwer Academic Publishers. Printed in the Netherlands.
Economic Equilibrium: Optimality and Price
, B. CORNET
and R. TOURKY
Department of Economics, Purdue University, West Lafayette, IN 47907–1310, USA.
CERMSEM, Maison des Sciences Economiques, Université Paris I, 106–112 Boulevard de
l’Hopital, 75645 Paris Cedex 13, France. E-mail: email@example.com
Department of Economics, University of Melbourne, Melbourne, VIC 3010, Australia.
(Received 10 August 2001; accepted 15 December 2001)
Abstract. Mathematical economics has a long history and covers many interdisciplinary areas
between mathematics and economics. At its center lies the theory of market equilibrium. The purpose
of this expository article is to introduce mathematicians to price decentralization in general equilib-
rium theory. In particular, it concentrates on the role of positivity in the theory of convex economic
analysis and the role of normal cones in the theory of non-convex economies.
AMS Classiﬁcation: 91, 46, 47
Key words: equilibrium, Pareto optimum, supporting price, properness, marginal cost pricing, vector
lattice, ordered vector space, Riesz–Kantorovich formula, normal cone, separation theorem
1. A Historical Survey
General equilibrium theory models the interaction of all economic agents in all
markets. Classically, it is assumed that this interaction is not strategic and that all
agents respond to linear price systems, which at equilibrium summarize the inform-
ation concerning relative scarcities and locally approximate the possibly non-linear
primitive data of the economy.
Advances in the theory of general equilibrium have gone hand-in-hand with the
study of the existence of at least one equilibrium price system. This is not surprising
since the existence problem was far more involved than what many economists had
anticipated in the past. With complexity came the need for rigor and rigor lead to
a better understanding of not only the existence problem but also the model as a
The purpose of this section is to informally and summarily trace the evolution
of the general equilibrium model from Léon Walras’ system of production and
exchange equations to the ‘state of the art’ model with inﬁnitely many commodities
and a ﬁnite number of consumers and producers.