Existence and uniqueness of Cournot equilibrium for an oligopoly under linear and nonlinear demand: a channel model for one manufacturer-two retailers with substitutes

Abstract

This paper considers the margin, price, and quantity decisions made by one manufacturer and two retailers in a two-echelon supply chain under Cournot competition. The manufacturer produces two types of substitutable products and sells one product to each retailer. The retailers carry one type of product, which is sold to the customer. The objective of this paper is to show existence and uniqueness of Cournot equilibrium for decisions made by one manufacturer and two retailers and to study the implications of these decisions through sensitivity analysis. This paper makes contributions to the literature in three ways. First, it extends on the literature by modeling an oligopoly of one manufacturer and two retailers selling differentiated products and acting in Cournot competition. Second, this paper demonstrates the sufficient conditions for existence and uniqueness of Cournot equilibrium in a two-echelon supply chain with linear and nonlinear demand. Third, this paper has three main findings: (1) the retailer with the lower price effect or lower price elasticity of demand has a higher margin, price, quantity demanded, and profit; (2) all players in the supply chain prefer to sell products with a high cross price effect or high cross price elasticity of demand; and (3) the manufacturer will always earn the most profit among the three members of the supply chain.