Positivity 6: 275–296, 2002.
© 2002 Kluwer Academic Publishers. Printed in the Netherlands.
The Marginal Pricing Rule in Economies with
Inﬁnitely Many Commodities
CERMSEM, Maison des Sciences Economiques, Université Paris I, 106–112 Boulevard de
l’Hopital, 75645 Paris Cedex 13, France (E-mail: email@example.com)
(Received 1 June 2000; accepted 16 August 2001)
Abstract. Clarke’s normal cone appears as the right tool to deﬁne the marginal pricing rule in
ﬁnite dimensional commodity space since it allows to consider in the same framework convex,
smooth as well as nonsmooth nonconvex production sets. Furthermore it has nice continuity and
convexity properties. But it is not well adapted for economies with inﬁnitely many commodities
since it does satisfy minimal continuity properties. In this paper, we propose an alternative deﬁnition
of the marginal pricing rule. It allows us to prove the second welfare theorem and the existence of
marginal pricing equilibria for economies with several producers under assumptions similar to the
one used for economies with a ﬁnite set of commodities. Our approach is sufﬁciently general to take
into account the convex and the smooth cases for which our deﬁnition of the marginal pricing rule
coincides with the one given by the Clarke’s normal cone or the normal cone of convex analysis.
AMS Classiﬁcation: 90A11, 90A14
Key words: General equilibrium, increasing returns, inﬁnitely many commodities, marginal pricing
The marginal cost pricing rule was introduced in the thirties to obtain sufﬁcient
conditions for the Pareto optimal allocation. To quote Hotelling , an optimum
of welfare “corresponds to the sale of everything at marginal cost.” When the
boundary of the production set is smooth and the cost function is well deﬁned
and differentiable, a price vector satisﬁes the marginal cost pricing rule for a given
production plan if it belongs to the unique outward normal half line to the produc-
tion set at the production plan (see ). This means that the relative prices must
be equal either to the marginal rates of transformation or to the marginal rates of
substitution. This also can be interpreted by saying that the producer fulﬁlls a ﬁrst-
order necessary condition for proﬁt maximization.
In a seminal paper, Guesnerie  proves the second welfare theorem in a
general equilibrium framework without smoothness or convexity assumptions on
the production sets. Considering nonsmooth production sets is relevant since, for
example, ﬁxed costs lead to nonsmooth nonconvex production sets or the sum of