Abstract
Motivation Pricing decisions are often made when market information is still poor. While modern pricing analytics aid firms to infer the distribution of the stochastic demand that they are facing, data-driven price optimization methods are often impractical or incomplete if not coupled with testable theoretical predictions. 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. Academic/practical relevance 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. Methodology 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. Results 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. Managerial implications Our findings complement data-driven decisions regarding price optimization and provide a systematic framework useful for making theoretical predictions in advance of market movements.
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Notes
The two-tier market model is an abstraction to capture the complexity of current markets. If we eliminate the second stage and assume that the supplier sells directly to the consumers, then our results still apply.
Under these assumptions, i.e., if the supplier’s capacity exceeds potential demand and if the retailers are symmetric, then the single price scheme is optimal among a wide range of possible pricing mechanisms (Harris and Raviv 1981; Riley and Zeckhauser 1983). In addition, the symmetry of the retailers allows the study of purely competitive aspects which is not possible if retailers are heterogeneous Tyagi (1999).
While our results directly apply to the more general demand function of López and Vives (2019), we stick to the linear model for expositional purposes. Except from the fact that linear markets have been consistently in the spotlight of economic research both due to their tractability and their accurate modeling of real situations, the study of the linear model is also technically motivated by Cohen et al. (2016) who demonstrate that when information about demand is limited, firms may act efficiently as if demand is linear. For practical purposes, this yields a simple and low regret pricing rule and provides a motivation to study linear markets in a systematic way.
To ease the exposition, we restrict to \(n=2\) retailers. As we show in Sect. 3.3, our results admit a straightforward generalization to arbitrary number n of symmetric retailers.
We will refer to \(\alpha \) throughout as the demand level. However, based on Eq. (2), \(\alpha \) is also known as the choke or reservation price. Since, these constants are equivalent up to some transformation in our model, this should cause no confusion.
Formally, this case contradicts the assumption that F is continuous or non-atomic. It is only allowed to avoid unnecessary notation and should cause no confusion.
Technically, these are perfect Bayes-Nash equilibria, since the supplier has a belief about the retailers’ types, i.e., their willingness-to-pay his price, that depends on the value of the stochastic demand parameter \(\alpha \).
In this literature, the MRD function is known as the mean residual life function due to its origins in reliability applications.
Since the DGMRD property is satisfied by a very broad class of distributions, see Banciu and Mirchandani (2013), Kocabıyıkoğlu and Popescu (2011) and Leonardos and Melolidakis (2018), we do not consider this as a significant restriction. Still, since it is sufficient (together with finiteness of the second moment) but not necessary for the existence of a unique optimal price, the analysis naturally applies to any other distribution that guarantees equilibrium existence and uniqueness.
In reliability applications, the MRD-order is commonly known as the mean residual life (MRL)-order, Shaked and Shanthikumar (2007).
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Acknowledgements
Stefanos Leonardos gratefully acknowledges support by the Alexander S. Onassis Public Benefit Foundation (PhD Scholarship) and partial support by NRF 2018 Fellowship, National Research Foundation Singapore (SG) NRF-NRFF2018-07.
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Leonardos, S., Melolidakis, C. & Koki, C. Monopoly pricing in vertical markets with demand uncertainty. Ann Oper Res 315, 1291–1318 (2022). https://doi.org/10.1007/s10479-021-04067-3
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DOI: https://doi.org/10.1007/s10479-021-04067-3
Keywords
- Monopoly pricing
- Revenue maximization
- Demand uncertainty
- Pricing analytics
- Comparative statics
- Stochastic orders
- Unimodality