# On the Equilibrium Uniqueness in Cournot Competition with Demand Uncertainty

## TL;DR

This paper establishes new sufficient conditions for the uniqueness of equilibrium in a Cournot model with demand uncertainty, specifically under the DMRD and IGFR properties, extending previous results and confirming conjectures.

## Contribution

It provides the first comprehensive conditions based on demand distribution properties that guarantee equilibrium uniqueness in uncertain demand Cournot models.

## Key findings

- Equilibrium is unique if demand intercept distribution has DMRD or IGFR properties.
- DMRD implies log-concavity of expected profits per unit.
- Answers Lagerl"of's conjecture that such conditions are not necessary.

## Abstract

We revisit the linear Cournot model with uncertain demand that is studied in Lagerl\"of (2006)* and provide sufficient conditions for equilibrium uniqueness that complement the existing results. We show that if the distribution of the demand intercept has the decreasing mean residual demand (DMRD) or the increasing generalized failure rate (IGFR) property, then uniqueness of equilibrium is guaranteed. The DMRD condition implies log-concavity of the expected profits per unit of output without additional assumptions on the existence or the shape of the density of the demand intercept and, hence, answers in the affirmative the conjecture of Lagerl\"of (2006)* that such conditions may not be necessary. *Johan Lagerl\"of, Equilibrium uniqueness in a Cournot model with demand uncertainty. The B.E. Journal in Theoretical Economics, Vol. 6: Iss 1. (Topics), Article 19:1--6, 2006.