Positivity 6: 205–241, 2002. © 2002 Kluwer Academic Publishers. Printed in the Netherlands. 205 Economic Equilibrium: Optimality and Price Decentralization C.D. ALIPRANTIS 1 , B. CORNET 2 and R. TOURKY 3 1 Department of Economics, Purdue University, West Lafayette, IN 47907–1310, USA. E-mail: aliprantis@mgmt.purdue.edu; 2 CERMSEM, Maison des Sciences Economiques, Université Paris I, 106–112 Boulevard de l’Hopital, 75645 Paris Cedex 13, France. E-mail: cornet@univ-paris1.fr 3 Department of Economics, University of Melbourne, Melbourne, VIC 3010, Australia. E-mail: rtourky@unimelb.edu.au (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 whole. 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.