Construction of simply connected open sets by positively homogeneous functions and its importance in the Marshallian consumer surplus analysis

Abstract

Two conditions assuring path independency in any line integral are that the integration domain be a simply connected open set and that the integrand have a symmetric Jacobain matrix in that domain. This paper provides a rigorous account of how a path-dependent line integral can retain path independency in a subset of the original domain. The implicit function theorem, intuitively, proves to be a key avenues toward the establishment of our main result when a positively homogeneous function defines the subset. The Poincare` lemma is used to derive a general proof. An application is made to the consumer surplus analysis in which such a technical concern is commonly ignored.