需求不确定下垂直市场的垄断定价
Monopoly Pricing in Vertical Markets with Demand Uncertainty
摘要：
定价决策往往是在市场信息仍然贫乏的情况下作出的。反过来，现有的理论模型往往原因的最优价格变化的市场特征的反应，而不利用所有可用的信息的需求分布。我们的目标是发展一种理论，用于在关于相应需求分布的各种形式的知识下，优化和系统地比较同一市场的不同实例之间的价格。我们重新审视了在一个垂直市场中，在需求不确定性下的垄断定价的经典问题，其中上游供应商和下游任意对称零售商之间存在多种形式的竞争。在所有情况下，需求的不确定性落在供应商身上，供应商首先采取行动，在零售商观察到已实现的需求并下订单之前制定统一的价格。我们的主要方法的贡献是，我们表示的平均剩余需求（MRD）的需求分布函数的预期需求的价格弹性。这导致了一个封闭的形式表征的点的单一的弹性，最大限度地提高供应商的利润和推导出一个温和的单峰条件供应商的目标函数，概括了广泛使用的增加广义故障率（IGFR）的条件。一个直接的含义是，不同市场之间的最优价格可以排序，如果市场可以随机排序根据他们的MRD功能或等价的，他们的弹性。利用上述内容，我们开发了一个系统的框架，通过丰富的随机订单理论来比较不同市场实例之间的最优价格。这导致了比较静态学，挑战了先前建立的关于市场规模、需求变化和需求变化对垄断价格的影响的经济学见解。
英文摘要：
Pricing decisions are often made when market information is still poor. In turn, existing theoretical models often reason about the response of optimal prices to changing market characteristics without exploiting all available information about the demand distribution. Our aim is to develop a theory for the optimization and systematic comparison of prices between different instances of the same market under various forms of knowledge about the corresponding demand distributions. We revisit the classic problem of monopoly pricing under demand uncertainty in a vertical market with an upstream supplier and multiple forms of downstream competition between arbitrary symmetric retailers. In all cases, demand uncertainty falls to the supplier who acts first and sets a uniform price before the retailers observe the realized demand and place their orders. Our main methodological contribution is that we express the price elasticity of expected demand in terms of the mean residual demand (MRD) function of the demand distribution. This leads to a closed form characterization of the points of unitary elasticity that maximize the supplier's profits and the derivation of a mild unimodality condition for the supplier's objective function that generalizes the widely used increasing generalized failure rate (IGFR) condition. A direct implication is that optimal prices between different markets can be ordered if the markets can be stochastically ordered according to their MRD functions or equivalently, their elasticities. Using the above, we develop a systematic framework to compare optimal prices between different market instances via the rich theory of stochastic orders. This leads to comparative statics that challenge previously established economic insights about the effects of market size, demand transformations and demand variability on monopolistic prices.
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