Econ Papers Daily / 期刊论文
非正则分布、虚拟价值与垄断问题
Non‐Regular Distributions, Virtual Values and Monopoly Problems
摘要
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.
中文摘要
我们重新审视了垄断模型中用于确保最优价格唯一性的标准正则性假设，即虚拟价值函数单调的条件。这一条件通常通过强分布假设（如对数凹性）来施加，但这些假设限制了模型的灵活性，并隐含地约束了比较静态的方向。我们证明，这些假设可以在不牺牲可处理性或唯一性的前提下被放宽。本文的主要贡献在于引入“局部正则性”这一更弱的条件，该条件仅要求虚拟价值函数在边际收入变为非负的阈值之后保持单调。我们证明，许多常见的非正则单峰分布满足这一局部条件。这一视角的转变使得标准的一阶方法仍然有效，并扩大了应用产业组织中可处理的需求基元集合。其含义在于，当风险率在相关定价范围内不单调时，比较静态的方向不必遵循对数凹性所施加的路径。该框架还减少了在建模重尾需求时进行“熨平”处理的必要性。