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Pricing with markups in industries with increasing marginal costs

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Abstract

We study a game that models a market in which heterogeneous producers of perfect substitutes make pricing decisions in a first stage, followed by consumers that select a producer that sells at lowest price. As opposed to Cournot or Bertrand competition, producers select prices using a supply function that maps prices to production levels. Solutions of this type of models are normally referred to as supply function equilibria. We consider a market where producers’ convex costs functions are proportional to each other, depending on the efficiency of each particular producer. We provide necessary and sufficient conditions for the existence of an equilibrium that uses simple supply functions that replicate the cost structure. We then specialize the model to monomial cost functions with exponent \(q>0\), which allows us to reinterpret the simple supply functions as a markup applied to the production cost. We prove that an equilibrium for the markups exists if and only if the number of producers in the market is strictly larger than \(1+q\), and if an equilibrium exists, it is unique. The main result for monomials is that the equilibrium nearly minimizes the total production cost when the market is competitive. The result holds because when there is enough competition, markups are bounded, thus preventing prices to be significantly distorted from costs. Focusing on the case of linear unit-cost functions on the production quantities, we characterize the equilibrium accurately and refine the previous result to establish an almost tight bound on the worst-case inefficiency of equilibria. Finally, we derive explicitly the producers’ best response for series-parallel networks with linear unit-cost functions, extending our previous result to more general topologies. We prove that a unique equilibrium exists if and only if the network that captures the market structure is 3-edge-connected. For non-series-parallel markets, we provide an example that does not admit an equilibrium on markups.

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Notes

  1. This is the case, among others, of the Chilean system, which operates with “audited costs” (a particular case of what is known in the literature as cost-based bids). In this system, the central dispatcher may audit firms that submit non-credible supply functions. Basically, the regulator knows the shape of the cost function (because he knows the technology used) but not the exact values. This could be the case, for example, because the regulator does not know the private contracts between the firm and its suppliers (coal, fuel, etc.).

  2. In fact, Klemperer and Meyer [26] already noted that in the case of deterministic demand an equilibrium is supported by supply functions that have the right value and slope at the right price.

  3. Note that because cost functions are all monomials of the same degree, the vector \(x^{{\textsc {NE}}}\) also represents the market shares that correspond to the solution that minimize the payments [16]. This solution would correspond to the case of a single buyer instead of the situation of perfect competition that we consider in this paper.

  4. Equation (6.2) matches that used for electrical circuits to compute the equivalent resistance when placing resistors in series and parallel. Ohm’s law, \(\textit{Voltage} = \textit{Current} \cdot \textit{Resistance}\), is analogous to the price function \(p_a = x_a R_a\). Although the equations describing both systems are identical, the difference is that we impose a nonnegativity restriction on flows, whereas in electricity networks this is not needed. It is precisely those restrictions that complicate the analysis of a general network as we will discuss in Sect. 6.4.

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Acknowledgments

This research was partially funded by the Millenium Nucleus Information and Coordination in Networks ICM/FIC P10-024F, the Instituto Sistemas Complejos de Ingeniería at Universidad de Chile, by Fondecyt Chile grant 1130671, by the Center for International Business Education and Research at Columbia University, and by Conicet Argentina grant Resolución 4541/12. The authors wish to thank the associate editor and an anonymous referee for their constructive comments, Roberto Cominetti, Andy Philpott, and Gabriel Weintraub for stimulating discussions, and the participants of seminars at Columbia Business School, the School of Industrial and Systems Engineering of Georgia Tech, Ecole Polytechnique Fédéral de Lausanne, IESE Business School, and Universidad de Chile, for their questions, comments and suggestions.

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Correspondence to Nicolás E. Stier-Moses.

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Correa, J.R., Figueroa, N., Lederman, R. et al. Pricing with markups in industries with increasing marginal costs. Math. Program. 146, 143–184 (2014). https://doi.org/10.1007/s10107-013-0682-8

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  • DOI: https://doi.org/10.1007/s10107-013-0682-8

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