Duality, Net Supply, and the Directional
Mauricio Benegas, Emerson Marinho
CAEN - UFC
July 21, 2008
This paper considers alternative proofs of the Hotelling´s Lemma
and Shephard´s Lemma using new relations of duality in the theory
of the production. The proofs proposals possess two advantages, rela-
tively to the usual demonstrations, which are, not the use of the known
Enveolpe Theorem and the validity of the demonstration for interior
points to the production set
Keywords: Duality, Hotelling´s Lemma, Shephard ´s Lemma.
Este artigo explora técnicas alternativas para demonstrar os Lemas
de Hotelling e de Shephard são fazer uso do Teorema do Envelope o
que permite que se obtenha sistemas de oferta de produtos e demanda
de insumos que sejam técnicamente ineficientes. Adicionalmente é
demonstrado que o Teorema da Dualidade de Shephard constitue caso
particular da dualidade entre a função lucro e a função distância di-
Palavras-chave: Dualidade, Lema de Shephard, Lema de Hotelling.
JEL: D24, C02, C61
Duality in the microeconomic theory of production was first investigated in
the work of Shephard (1953, 1970, 1973) — the precursor of the axiomatic
approach to production theory1. The results obtained by this author and
later improved by McFadden (1978), Blackorby & Donaldson (1980) and
Färe & Primont (1995), enabled important applications in production the-
ory, notably with regard to obtaining functions of product supply and input
demand, through simples derivation of functions, values such as the profit
function, without the need to undertake and optimization process.
The vast majority of results obtained by these authors, however, make
extensive use of the Envelope Theorem, which, to a certain extent, requires
that one know the optimization process underlying the productive process.
Furthermore, it should be pointed out that the use of the Envelope The-
orem implicitly requires that the choice of product and input vectors be
technical and allocativ