From ordients to optimization: Substitution effects without differentiability

This paper introduces the concept of ordient for binary relations (preferences), a relative of the concept of gradient for functions (utilities). The main motivation for this study is to replace the binary relation at the center stage of economic analysis, rather than its representation (whenever it exists). Moreover, ordients have a natural economic interpretation as marginal rates of substitution. Some examples of ordientable binary relations include the lexicographic order, binary relations resulting from the sequential applications of multiple rationales or binary relations with differentiable representations. We characterize the constrained maxima of binary relations through ordients and provide an implicit function theorem and an envelope theorem.