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

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 varying 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.