Optimal collusion with limited liability

Billette de Villemeur, Etienne and Flochel, Laurent and Versaevel, Bruno (2012): Optimal collusion with limited liability.

Abstract

Collusion sustainability depends on firms' aptitude to impose sufficiently severe punishments in case of deviation from the collusive rule. We extend results from the literature on optimal collusion by investigating the role of limited liability. We examine all situations in which either structural conditions (demand and technology), financial considerations (a profitability target), or institutional circumstances (a regulation) set a lower bound, possibly negative, to firms' profits. For a large class of repeated games with discounting, we show that, absent participation and limited liability constraints, there exists a unique optimal penal code. It commands a severe single-period punishment immediately after a firm deviates from the collusive stage-game strategy. When either the participation constraint or the limited liability constraint bind, there exists an infinity of multi-period punishment paths that permit firms to implement the optimal collusive strategy. The usual front-loading scheme is only a specific case and an optimal punishment profile can take the form of a price asymmetric cycle. We characterize the situations in which a longer punishment does not perform as a perfect substitute for more immediate severity. In this case the lowest discount factor that permits collusion is strictly higher than without the limited liability constraint, which hinders collusion.

• Item Type: MPRA Paper
• Language: English
• Keywords: Collusion; Oligopoly; Limited Liability
• Subjects: L - Industrial Organization > L1 - Market Structure, Firm Strategy, and Market Performance > L13 - Oligopoly and Other Imperfect Markets; D - Microeconomics > D4 - Market Structure, Pricing, and Design > D43 - Oligopoly and Other Forms of Market Imperfection; C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C72 - Noncooperative Games

References

Abreu, D. (1986). "Extremal equilibria of oligopolistic supergames", Journal of Economic Theory, 39(1), 191-225.

Abreu, D. (1988). "On the theory of infinitely repeated games with discounting", Econometrica, 56(2), 383-396.

Athey, S., and Bagwell, K. (2001). "Optimal collusion with private information", RAND Journal of Economics, 32(2), 428-465.

Athey, S., and Bagwell, K., and Sanchirico, C. (2004). "Collusion and price rigidity", Review of Economic Studies, 71(2), 317-349.

Athey, S., and Bagwell, K. (2008). "Collusion with persistent cost shocks", Econometrica, 76(3), 493-540.

Bagwell, K., and Staiger, R. W. (1997). "Collusion over the business cycle", RAND Journal of Economics, 28(1), 82-106.

Baker, M., Stein, J. C., and Wurgler, J. (2003). "When does the market matter? Stock prices and the investment of equity-dependent firms", Quarterly Journal of Economics, 118(3), 969-1005.

Beviá, C., Corchón, L. C., and Yasuda, Y. (2011). "Oligopolistic equilibrium and financial constraints", Universidad Carlos III de Madrid, Working Paper 11-10.

Compte, O, Jenny, F., and Rey, P. (2002). "Capacity constraints, mergers and collusion", European Economic Review, 46(1), 1-29.

Deneckere, R. (1983). "Duopoly supergames with product differentiation", Economics Letters, 11(1), 37-42. Deneckere, R. (1984). "Corrigenda", Economics Letters, 15, 385-87.

Eckert, A. (2002). "Retail price cycles and response asymmetry", Canadian Journal of Economics, 35 (1), 52-77.

Eckert, A., and West, D. S. (2004). "Retail gasoline price cycles across spatially dispersed gasoline stations", Journal of Law and Economics, 47(1), 245-274.

Friedman, J. W. (1971). "A non-cooperative equilibrium for supergames", Review of Economic Studies, 38(1), 1-12.

Fudenberg, D., and Maskin, E. (1986). "The Folk theorem in repeated games with discounting or with incomplete information", Econometrica, 54(3), 533-554.

Fudenberg, D., and Tirole, J. (1991). Game Theory, Cambridge: MIT Press.

Gagné, R., van Norden, S., and Versaevel, B. (2006). "Testing optimal punishment mechanisms under price regulation: the case of the retail market for gasoline", GATE Working Paper 06-11, 28 p.

Green, E. J., and Porter, R. H. (1984). "Noncooperative collusion under imperfect price information", Econometrica, 52(1), 87-100.

Häckner, J. (1996). "Optimal symmetric punishments in a Bertrand differentiated products duopoly", International Journal of Industrial Organization, 14(5), 611-630.

Häckner, J. (2000). "A note on price and quantity competition in differentiated oligopolies", Journal of Economic Theory, 77(1), 181-191.

Haltiwanger, J., and Harrington, J. E. (1991). "The impact of cyclical demand movements on collusive behavior", RAND Journal of Economics, 22(1), 89-106.

Houde, J.-F. (2008). "Entry and exit in a price regulated industry: gasoline retailing in Quebec", University of Wisconsin-Madison, mimeo.

Houde, J.-F. (2010). "Gasoline markets", in: S. N. Durlauf and L. E. Blume (Eds.), The New Palgrave Dictionary of Economics Online, Palgrave Macmillan, forthcoming.

Lambertini, L., and Sasaki, D. (2001). "Marginal costs and collusive sustainability", Economics Letters, 72(1), 117-120.

Lambertini, L., and Sasaki, D. (2002). "Non-negative quantity constraints and the duration of punishment", Japanese Economic Review, 53(1), 77-93.

Lambson, V. E. (1987). "Optimal penal codes in price-setting supergames with capacity constraints", Review of Economic Studies, 54(3), 385-397.

Lambson, V. E. (1994). "Some results on optimal penal codes in asymmetric Bertrand supergames", Journal of Economic Theory, 62(2), 444-468.

Levenstein, M. C., and Suslow, V. Y. (2006). "What determines cartel success?", Journal of Economic Literature, 44(1), 43-95.

Miklós-Thal, J. (2011). "Optimal collusion under cost asymmetry", Economic Theory, 46(1), 99-125.

Motta, M. (2004). Competition Policy: Theory and Practice, Cambridge University Press.

Noel, M. D. (2006). "Edgeworth price cycles, cost-based pricing, and sticky pricing in retail Gasoline markets", Review of Economics and Statistics, 89(2), 324-334.

Noel, M. D. (2007). "Edgeworth price cycles: Evidence from the Toronto retail gasoline market", Journal of Industrial Economics, 55(1), 69-92.

Osterdal, L. P. (2003). "A note on the stability of collusion in differentiated oligopolies", Research in Economics, 57(1), 53-64.

Porter, R. H. (1983). "Optimal Cartel Trigger-Price Strategies", Journal of Economic Theory, 29(2), 313-338.

Rotemberg, J. J., and Saloner, G. (1986), "A supergame-theoretic model of price wars during booms", American Economic Review, 76(3), 390-407.

Rothschild, R. (1999). "Cartel stability when costs are heterogeneous", International Journal of Industrial Organization, 17(5), 717-734.

Symeonidis, G. (2003). "In which industries is collusion more likely? Evidence from the UK", Journal of Industrial Economics, 51(1), 45-74.

Vasconcelos, H. (2005). "Tacit collusion, cost asymmetries and mergers", RAND Journal of Economics, 36(1), 39-62.

Wernerfelt, B. (1989). "Tacit collusion in differentiated Cournot games", Economics Letters, 29(4), 303-306.

Yasuda, Y. (2009). "The theory of collusion under financial constraints", National Graduate Institute for Policy Studies (GRIPS), mimeo.