Non‐Regular Distributions, Virtual Values and Monopoly Problems

ABSTRACT We revisit the standard regularity assumption used to ensure uniqueness of the optimal price in monopoly models, namely that the Virtual Value (VV) function is monotone. This condition is typically imposed through strong distributional assumptions, such as log‐concavity, that restrict the model's flexibility and implicitly discipline the sign of comparative statics. We show that these assumptions can be relaxed without compromising tractability or uniqueness. Our main contribution is to introduce ‐Regularity, a weaker condition that requires monotonicity of the VV only beyond the threshold where marginal revenue becomes nonnegative. We demonstrate that many common non‐regular, unimodal distributions satisfy this local condition. This shift in focus allows standard first‐order methods to remain valid, and it enlarges the set of tractable demand primitives in applied IO. As an implication, when hazard rates are not monotone over the relevant pricing range, comparative statics need not follow the direction imposed by log‐concavity. The framework also reduces the need for ironing when modeling with heavy‐tailed demand.