Comparative Statics via Stochastic Orderings in a Two-Echelon Market with Upstream Demand Uncertainty

TL;DR

This paper analyzes a two-echelon supply chain with demand uncertainty using stochastic orderings, providing a clear equilibrium characterization and sensitivity analysis of prices, quantities, and efficiency metrics under demand distribution variations.

Contribution

It introduces a novel equilibrium analysis framework for supply chains with demand uncertainty using the DGMRL property and stochastic orderings, enabling transparent comparative statics.

Findings

Equilibrium strategies are characterized by a fixed point of the MRL function.

Demand distribution parameters significantly impact prices and quantities.

Supply chain efficiency varies with demand uncertainty, affecting profits and Price of Anarchy.

Abstract

We revisit the classic Cournot model and extend it to a two-echelon supply chain with an upstream supplier who operates under demand uncertainty and multiple downstream retailers who compete over quantity. The supplier's belief about retail demand is modeled via a continuous probability distribution function F. If F has the decreasing generalized mean residual life (DGMRL) property, then the supplier's optimal pricing policy exists and is the unique fixed point of the mean residual life (MRL) function. This closed form representation of the supplier's equilibrium strategy facilitates a transparent comparative statics and sensitivity analysis. We utilize the theory of stochastic orderings to study the response of the equilibrium fundamentals - wholesale price, retail price and quantity - to different demand distribution parameters. We examine supply chain performance, in terms of the distribution of profits, supply chain efficiency, in terms of the Price of Anarchy, and complement our findings with numerical results.