S-Convexity and Gross Substitutability

47 Pages Posted: 31 Mar 2020 Last revised: 12 Nov 2021

See all articles by Xin Chen

Xin Chen

Georgia Institute of Technology - H. Milton Stewart School of Industrial and Systems Engineering

Menglong Li

City University of Hong Kong (CityU) - Department of Management Sciences

Date Written: March 5, 2020

Abstract

We propose a new concept of S-convex functions (and its variant SSQS-convex functions) to study substitute structures in economics and operations models with continuous variables. This class of functions is a significant generalization of M-natural-convex functions (a key concept in discrete convex analysis) over continuous spaces. We develop a host of fundamental properties and characterizations of S-convex functions including various preservation properties, conjugate relationship with submodular and convex functions, and characterizations using Hessians. For a divisible Fisher market with quasilinear utilities, we show that the utility function satisfies gross substitutability if and only if it is S-concave under mild regularity conditions. In a parametric maximization model with a box constraint, we show that the set of optimal solutions is nonincreasing in the parameters if the objective function is (SSQS-)S-concave. Furthermore, we prove that S-convexity is necessary for the property of nonincreasing optimal solutions under some conditions. Our monotonicity result is applied to analyze two notable inventory models. In a random yield inventory model where a buyer procures a single product from multiple unreliable suppliers to fulfill its random demand, we provide conditions under which the ordering quantity vector is nonincreasing in the initial inventory. For a classical multi-product inventory model, we show that S-convexity leads to two desired decreasing properties, which includes several results in the literature as special cases and significantly simplifies their analysis

Keywords: S-convexity, M-natural-convexity, gross substitutability, inventory models

Suggested Citation

Chen, Xin and Li, Menglong, S-Convexity and Gross Substitutability (March 5, 2020). Available at SSRN: https://ssrn.com/abstract=3549632 or http://dx.doi.org/10.2139/ssrn.3549632

Xin Chen

Georgia Institute of Technology - H. Milton Stewart School of Industrial and Systems Engineering ( email )

Menglong Li (Contact Author)

City University of Hong Kong (CityU) - Department of Management Sciences ( email )

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