Skip to main content
Log in

Pricing with markups in industries with increasing marginal costs

  • Full Length Paper
  • Series A
  • Published:
Mathematical Programming Submit manuscript


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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others


  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.


  1. Acemoglu, D., Bimpikis, K., Ozdaglar, A.: Price and capacity competition. Games Econ. Behav. 66(1), 1–26 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  2. Acemoglu, D., Ozdaglar, A.: Competition and efficiency in congested markets. Math. Oper. Res. 32(1), 1–31 (2007)

    Google Scholar 

  3. Acemoglu, D., Ozdaglar, A.: Competition in parallel-serial networks. IEEE J. Sel. Areas Commun. 25(6), 1180–1192 (2007)

    Google Scholar 

  4. Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice-Hall, Englewood Cliffs (1993)

    MATH  Google Scholar 

  5. Akgün, U.: Mergers with supply function. J. Ind. Econ. 52(4), 535–545 (2004)

    Article  Google Scholar 

  6. Anderson, E.J., Hu, X.: Finding supply function equilibria with asymmetric firms. Oper. Res. 56(3), 697–711 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Anderson, E.J., Philpott, A.B.: Using supply functions for offering generation into an electricity market. Oper. Res. 50(3), 477–489 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  8. Baldick, R., Grant, R., Kahn, E.: Theory and application of linear supply function equilibrium in electricity markets. J. Regul. Econ. 25(2), 143–167 (2004)

    Article  Google Scholar 

  9. Baldick, R., Hogan, W.: Stability of supply function equilibria: Implications for daily versus hourly bids in a poolco market. J. Regul. Econ. 30(2), 119–139 (2006)

    Article  Google Scholar 

  10. Beckmann, M.J., McGuire, C.B., Winsten, C.B.: Studies in the Economics of Transportation. Yale University Press, New Haven (1956)

    Google Scholar 

  11. Bowley, A.: The Mathematical Groundwork of the Economics. Oxford University Press, Oxford (1924)

    Google Scholar 

  12. Casadesus-Masanell, R., Nalebuff, B., Yoffie, D.B.: Competing complements. Harvard Business School Working Paper, No. 09-009 (2010)

  13. Chau, C.K., Sim, K.M.: The price of anarchy for non-atomic congestion games with symmetric cost maps and elastic demands. Oper. Res. Lett. 31(5), 327–334 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  14. Chawla, S., Roughgarden, T.: Bertrand competition in networks. In: Monien, B., Schroeder, U. (eds.) Algorithmic Game Theory, First International Symposium (SAGT 2008), volume 4997 of Lecture Notes in Computer Science. Springer (2008)

  15. Cournot, A.: Recherches sur les Principes Mathematiques de la Theorie des Richesses. Hachette, Paris (1838)

    Google Scholar 

  16. Dafermos, S.C., Sparrow, F.T.: The traffic assignment problem for a general network. J. Res. U.S Natl. Bureau Stand. 73B, 91–118 (1969)

    Article  MathSciNet  Google Scholar 

  17. Engel, E., Fischer, R., Galetovic, A.: Toll competition among congested roads. Top. Econ. Anal. Policy 4(1). Article 4 (2004)

    Google Scholar 

  18. Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics: A Foundation for Computer Science, 2nd edn. Addison-Wesley, Reading, MA (1994)

  19. Granot, D., Yin, S.: Competition and cooperation in decentralized push and pull assembly systems. Manage. Sci. 54(3), 733–747 (2008)

    Article  MATH  Google Scholar 

  20. Grossman, S.: Nash equilibrium and the industrial organisation of markets with large fixed costs. Econometrica 49, 1149–1172 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  21. Hendricks, K., McAfee, R.P.: A theory of bilateral oligopoly. Econ. Inq. 48(2), 391–414 (2010)

    Google Scholar 

  22. Jiang, L., Wang, Y.: Supplier competition in decentralized assembly systems with price-sensitive and uncertain demand. Manuf. Serv. Oper. Manage. 12(1), 93–101 (2010)

    Google Scholar 

  23. Johari, R., Tsitsiklis, J.N.: Network resource allocation and a congestion game. Math. Oper. Res. 29(3), 407–435 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  24. Johari, R., Tsitsiklis, J.N.: Parameterized supply function bidding: equilibrium and welfare. Oper. Res. 59(5), 1079–1089 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  25. Johari, R., Weintraub, G.Y., Van Roy, B.: Investment and market structure in industries with congestion. Oper. Res. 58(5), 1303–1317 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  26. Klemperer, P.D., Meyer, M.: Supply function equilibria in oligopoly under uncertainty. Econometrica 57(6), 1243–1277 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  27. Koutsoupias, E., Papadimitriou, C.H.: Worst-case equilibria. In: Meinel, C., Tison, S. (eds.) Proceedings of the 16th Annual Symposium on Theoretical Aspects of Computer Science (STACS), Trier, Germany, Volume 1563 of Lecture Notes in Computer Science, pp. 404–413. Springer, Heidelberg (1999)

  28. Kuleshov, V., Wilfong, G.: Cournot and bertrand competition in two-sided markets. Preprint (2011)

  29. Lederman, R.: Strategic models in supply network design. Ph. D. thesis, Graduate School of Business, Columbia University, New York, NY (2012)

  30. Mas-Colell, A., Whinston, M.D., Green, J.R.: Microeconomic Theory. Oxford University Press, New York (1995)

    MATH  Google Scholar 

  31. McAfee, R.P., Williams, M.A.: Horizontal mergers and antitrust policy. J. Ind. Econ. 40(2), 181–187 (1992)

    Article  Google Scholar 

  32. Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, Berlin (2006)

    MATH  Google Scholar 

  33. Papadimitriou, C., Valiant, G.: A new look at selfish routing. In: Chi-Chih Yao, A.(ed.) Innovations in Computer Science (ICS). Tsinghua University Press, Beijing, China (2010)

  34. Perry, M.K., Porter, R.H.: Oligopoly and the incentive for horizontal merger. Am. Econ. Rev. 75(1), 219–225 (1985)

    Google Scholar 

  35. Robson, A.: Implicit oligopolistic collusion is destroyed by uncertainty. Econ. Lett. 7, 144–148 (1981)

    Article  Google Scholar 

  36. Rudkevich, A., Duckworth, M., Rosen, R.: Modeling electricity pricing in a deregulated generation industry: the potential for oligopoly pricing in a poolco. Energy J. 19(3), 19–48 (1998)

    Article  Google Scholar 

  37. Sioshansi, R., Oren, S.: How good are supply function equilibrium models: an empirical analysis of the ERCOT balancing market. J. Regul. Econ. 31, 1–35 (2007)

    Article  Google Scholar 

  38. Turnbull, S.: Choosing duopoly solutions by consistent conjectures and by uncertainty. Econ. Lett. 13, 253–258 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  39. Vives, X.: Strategic supply function competition with private information. Econometrica 79(6), 1919–1966 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  40. Wang, Y.: Joint pricing-production decisions in supply chains of complementary products with uncertain demand. Oper. Res. 54(6), 1110–1127 (2006)

    Article  MATH  Google Scholar 

  41. Wichiensin, M., Bell, M.G.H., Yang, H.: Impact of congestion charging on the transit market: an inter-modal equilibrium model. Transp. Res. 41A(7), 703–713 (2007)

    Google Scholar 

  42. Wilson, R.: Auctions of shares. Q. J. Econ. 94, 675–689 (1979)

    Article  Google Scholar 

  43. Xiao, F., Yang, H., Han, D.: Competition and efficiency of private toll roads. Transp. Res. 41B(3), 292–308 (2007)

    Article  Google Scholar 

  44. Yang, S., Hajek, B.: Revenue and stability of a mechanism for efficient allocation of a divisible good. Working paper (2006)

Download references


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.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Nicolás E. Stier-Moses.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Correa, J.R., Figueroa, N., Lederman, R. et al. Pricing with markups in industries with increasing marginal costs. Math. Program. 146, 143–184 (2014).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


Mathematics Subject Classification