Abstract
We examine the myopic price changes based on the inverse elasticity pricing rule for a multiproduct firm. By myopic, we mean ignoring that elasticity likely changes with price and that marginal cost likely changes with quantity. Unlike with a single product firm, constant demand elasticities and marginal cost do not cause the inverse elasticity rule to yield the profit-maximizing price change. The price change will be too large if the related products are complements which are relatively inelastic and too small otherwise. Importantly, whether non-constant marginal cost causes too large or too small, a price change is independent of whether the related products are substitutes or complements. Increasing marginal costs always causes too large an increase and decreasing marginal costs always causes too small an increase. While not providing a full characterization, we partially identify the net effects of allowing elasticities, cross-price elasticities, and marginal cost to be non-constant.
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Reibstein and Gatignon (1984) also include uncertainty and Tirole (1988) expresses (2) in terms of the price–cost margin as follows: \(\frac{{p}_{i}-{c}_{i}}{{p}_{i}}=\frac{-1}{{\varepsilon }_{ii}}-\sum_{j\ne i}\left({p}_{j}-{c}_{j}\right)\left(\frac{{\varepsilon }_{ji}}{{\varepsilon }_{ii}}\right)\frac{{q}_{j}}{{{p}_{i}q}_{i}}\). The expression in (2) is sometimes also referred to as the Niehans formula (Simon and Fassnacht 2019).
We recognize that altering these assumptions will lead to even more complex influences on the difference between myopic and optimal pricing. Yet, as our primary objective is to show how poorly the multiple product rule of thumb works compared to the single product rule of thumb, we leave these complexities aside.
The other marginal cost effect might also be called the margin effect, since it pertains to the price-marginal cost margin of the related product, (\({p}_{j}-{c}_{j}\)).
Tirole (1988) uses the example of each product being produced by a separate division of a profit-maximizing firm. For substitutes, the divisions are de facto competitors and must be given incentives to raise price to eliminate externalities between them. They must be given incentives to lower price if the divisions produce complements.
The relative quantities effect can be zero if, by chance, \({\varepsilon }_{ji}={\varepsilon }_{ii}\) since this will leave the relative quantities unchanged. This can only happen for complementary goods.
The net effect could be zero in the extremely unlikely case that all marginal costs and elasticities are constant, and all related products are complements to the priced product and where \({\varepsilon }_{ji}={\varepsilon }_{ii}\) for all of them.
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We are grateful to Sophie Maussen and anonymous reviewers for helpful comments.
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The first draft of the manuscript was written by KF. Both authors have made substantial contributions to the analysis and interpretation of the results. Both authors have revised the manuscript critically for important intellectual content. Both authors have read and approved the final manuscript.
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Fjell, K., Heywood, J.S. Myopic use of the inverse elasticity pricing rule by a multiproduct firm. J Revenue Pricing Manag 23, 103–111 (2024). https://doi.org/10.1057/s41272-023-00436-8
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DOI: https://doi.org/10.1057/s41272-023-00436-8