Abstract
According to the Lerner rule, a firm’s profit-maximizing price under certainty can be characterized with just two parameters: marginal cost and the elasticity of demand. Salinger and Ampudia (Salinger, M. A., & Ampudia, M. (2011). Simple economics of the price-setting newsvendor problem. Managament Science, 57, 1996–1998.) showed that in the most basic version of the price-setting newsvendor (i.e., with no inventories or stock-out costs), the Lerner rule applies with suitable modifications to the definition of marginal cost and the elasticity of demand. This chapter extends that result to the more general version of the price-setting newsvendor problem that allows for stock-out costs and for unsold output to have some residual value as inventory. This extension suggests that the Lerner rule characterization can be a unifying framework for a wide variety of extensions to the price-setting newsvendor problem.
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As we discuss in more detail below, the elasticity of the average quantity sold with respect to price reflects a weighted average over demand states, with the probabilty of selling all output being the weight for the demand state in which consumers demand exactly the quantity produced.
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As is discussed in more detail below, both factors reflect specific assumptions about how price changes in conjunction with changes in output.
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There are other possible extensions. For example, we maintain the assumption that the newsvendor is risk neutral and therefore seeks to maximize expected profits. Other possible extensions include allowing for an objective function other than expected profit maximization (Arcelus et al. 2012; Wang et al. 2009; Choi and Chiu 2010; Yang et al. 2011) and uncertainty in costs (Tang et al. 2011).
- 5.
They could also reflect direct costs to the extent that the firm has an obligation to supply. For example, a store that advertises a good at particular price might have an obligation to satisfy demand in some way; and doing so might prove costly. Stock-out costs are likely to be particularly important in extending newsvendor analysis to oligopolistic industries. See Krishnan and Winter 2007. For an empirical estimate of stock-out costs, see Matsa 2011.
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“Constant” can refer to a single state of demand or an average across a distribution of states.
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Salinger and Ampudia refer to this as a “general stocking factor approach.” Because the stocking factor typically refers to a physical quantity, “quality transformation” is a better characterization of the approach.
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See also Raz and Porteus (2006) fractile approach, which is a discrete approximation to the generalized stocking factor approach.
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The assumption that ε is uniformly distributed is a consequence of Q(ε, p) being an inverse cumulative distribution function. It does not impose any particular form of the distribution function itself.
- 11.
While the Lerner rule under certainty and the extensions to uncertainty developed here apply to any firm provided that the demand parameters and uncertainty are understood as those facing the individual firm, explicitly modeling the link between market demand uncertainty and the uncertainty facing an individual firm would greatly extend the scope for practical application.
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Ampudia, M., Salinger, M.A. (2012). Inventories and Stock-out Costs in the Price-Setting Newsvendor: An Economic Interpretation. In: Choi, TM. (eds) Handbook of Newsvendor Problems. International Series in Operations Research & Management Science, vol 176. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3600-3_6
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