Sequential Multi-Product Price Competition in Supply Chain Networks

Abstract

We analyze a general model in which, at each echelon of the supply process, an arbitrary number of firms compete, offering one or multiple products to some or all of the firms at the next echelon, with firms at the most downstream echelon selling to the end consumer. At each echelon, the offered products are differentiated and the firms belonging to this echelon engage in price competition. The model assumes a general set of piece-wise linear consumer demand functions for all products (potentially) brought to the consumer market, where each product's demand volume may depend on the retail prices charged for all products; consumers' preferences over the various product/retailer combinations are general and asymmetric. Similarly the cost rates incurred by the firms at the most upstream echelon are general as well.

We initially study a two-echelon sequential oligopoly with competing suppliers, each selling multiple products through a pool of multiple competing retailers. We characterize the equilibrium behavior under linear price-only contracts. In the second stage, given wholesale prices selected in the first stage, all retailers simultaneously decide on their retail prices to maximize their total profits among all products of all suppliers they choose to do business with. In the first stage, the suppliers anticipate the retailers' responses and all suppliers simultaneously maximize their total profits from all channels by selecting the wholesale prices. We show that in this two-stage competition model, a subgame perfect Nash equilibrium always exists. Multiple subgame perfect equilibria may arise but, if so, all equilibria are equivalent in the sense of generating unique demands and profits for all firms. We subsequently generalize our results to supply chain models with an arbitrary set of echelons, and show how all equilibrium performance measures can be computed with an efficient recursive scheme. Moreover, we establish how changes in the structure of the supply chain network, or changes in the model parameters, in particular, exogenous cost rates, or intercept values in the demand functions, impact on the system-wide equilibrium. These comparative statics results allow for the quantification of cost pass-through effects and the measurement and characterization of the brand value of different retailers and suppliers.

Keywords: Stackelberg game, Bertrand competition, sequential oligopoly, multi-echelon, Linear Complementarity Problem

JEL Classification: C6, C72, D43, D58, L13