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Collusion in Differentiated Duopolies with Quadratic Costs
Abstract
The analysis of collusion in infinitely repeated duopoly games has generally assumed that marginal cost is constant, but this note uses quadratic costs (linear marginal costs) to compare the sustainability of collusion under Bertrand and Cournot duopoly with differentiated products. It is shown that when marginal costs are sufficiently increasing in output, then it is always easier to sustain collusion under Cournot duopoly than under Bertrand duopoly for any degree of product substitutability. Copyright Blackwell Publishers Ltd and the Board of Trustees of the Bulletin of Economic Research, 2006.
... Deneckere [2] pointed out the impact of product differentiation on collusion. Cost structures also play an important role when it comes to collusion (Collie [3]). In recent years, the effects of network externalities have received extensive attention; Song and Wang [4] first took network externalities into account in the framework of collusion stability with symmetric cost. ...
... Toshimitsu [5] demonstrated the conditions under which collusive behavior improves social welfare. In mostly related work, Choi and Lee [6] showed that if the network externality is strong (weak), the collusion of price (quantity) is more stable than quantity (price), which is different from the findings of Collie [3]. In this paper, we aim to take network externalities and cost asymmetry into account to analyze downstream collusive stability in a vertical market. ...
... Lambertini and Sasaki [10] derived the optimal punishments required to sustain collusion under Bertrand and Cournot duopoly with differentiated products. Collie [3] showed that collusion is more sustainable under Cournot duopoly than under Bertrand duopoly with quadratic costs for any degree of product substitutability. ...
This paper considers the collusive stability of downstream competition in a vertical market with network externalities and cost asymmetry. A dynamic collusion game is constructed, and backward induction is employed to solve the subgame perfect Nash equilibrium. We show that larger network externalities lead to less collusive incentive for an inefficient firm, while for an efficient firm, this depends on the efficiency gap. An increase in network externalities will destabilize the downstream collusion when the cost asymmetry is large and network externalities are relatively weak.
... Collusion impedes competition and hence is undesirable from the social planner's perspective who aims to maximize societal welfare. Umpteen studies have been undertaken for the study of collusion among firms from various perspectives such as product differentiation (Deneckere 1983;Wernerfelt 1989;Chang 1991;Ross 1992;Symeonidis 2002;Collie 2006), asymmetric firms (Harrington 1989), cheap talks (Miralles 2010), cost structures [constant costs (Gibbons 1992;Martin 2001;Shy 1996;Tirole 1988)], convex costs (Collie 2006;Weibull 2006), and technological advancements (Marjit et al. 2017). Among other things, collusion is a function of the number of firms in the market. ...
... Collusion impedes competition and hence is undesirable from the social planner's perspective who aims to maximize societal welfare. Umpteen studies have been undertaken for the study of collusion among firms from various perspectives such as product differentiation (Deneckere 1983;Wernerfelt 1989;Chang 1991;Ross 1992;Symeonidis 2002;Collie 2006), asymmetric firms (Harrington 1989), cheap talks (Miralles 2010), cost structures [constant costs (Gibbons 1992;Martin 2001;Shy 1996;Tirole 1988)], convex costs (Collie 2006;Weibull 2006), and technological advancements (Marjit et al. 2017). Among other things, collusion is a function of the number of firms in the market. ...
Conventionally, entry is thought to enhance welfare by enhancing competition and hence lowering prices and increasing the output. Contrary to the conventional wisdom, working with an n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document}-firm Cournot oligopoly set up and using the trigger strategies, we show that entry may or may not impact welfare. However, entry has the potential to alter the market structure from collusion to Cournot competition, and when it does so, there is a discontinuous rise in welfare.
... The issue of collusion has been addressed from various perspectives such as product differentiation (Deneckere, 1983;Wernerfelt, 1989;Chang, 1991;Ross, 1992;Symeodinis, 2002;and Collie, 2006), asymmetric firms (Harrington, 1989), and cheap talks (Miralles, 2010). In the context of homogeneous goods with constant marginal cost, Gibbons (1992), Martin (2001), Shy (1996), andTirole (1988)shows that the existence of collusion or Cournot competition as the subgame perfect equilibrium (SPE) in an infinitely repeated game is independent of the technology. ...
... Collie (2006) and Weibull (2006) consider convex cost to study conditions that lead firms to collude. WhileCollie (2006)shows that if marginal cost is sufficiently increasing, then for any degree of product substitutability collusion is more easily sustainable in a Cournot set up than in a Bertrand set up, Weibull (2006) generalizes the Bertrand model from linear cost to convex cost functions and hints that firms profits may be increasing in their production costs. The above papers have assumed convex costs but their focus and results significantly differ from ours. ...
In this paper we show that the increasing marginal cost assumption removes the infeasibility of market structurealteration that is present under the constant marginal cost assumption. Specifically, in an infinitely repeated game withincreasing marginal cost, we show that technological improvement has the potential to switch the market structurefrom collusion to Cournot generating additional welfare gains.
... The literature on cartel stability affected by factors intrinsic to the market structure, such as the degree of product differentiation (Deneckere 1983;Majerus 1988;Ross 1992; Albaek and Lambertini 1998), 1 the difference in costs (Rothschild 1999;Vasconcelos 2005;Collie 2006;Miklos-Thal 2011;Ciarreta and Gutiérrez-Hita 2012), the volatility of the demand system, etc., has been extensively explored. 2 However, one of the interests of the antitrust authorities is to investigate how the antitrust policies (external factor to the market structure) affect the stability of cartels. ...
This paper examines the effect of antitrust enforcement on the stability of the quantity-setting cartel at different levels of horizontal product differentiation. We derive an incentive compatibility constraint that allows us to examine the effect of antitrust enforcement on cartel stability at different levels of horizontal product differentiation. We verified that both the antitrust enforcement and the degree of product differentiation affect cartel stability. We found that the relationship between the degree of product differentiation and cartel stability is increasing—if the products are more homogeneous, the cartel is less stable. Next, we prove that antitrust enforcement is more efficient (that is, its impacts more intense) in destabilizing the cartel if its products are highly differentiated, i.e., the cartel under antitrust enforcement is more stable when its products tend to be homogeneous.
... Collie (2006) analyzed that when marginal costs sufficiently increase in output, it is always easier to sustain collusion under Cournot duopoly than under Bertrand duopoly for any degree of product substitutability. See also collusion references therein.3Lambertini ...
This paper analyzes the impacts of downstream CSR on upstream tacit collusion. Considering an infinitely repeated game with trigger strategy punishment, we find that how consumer‐oriented CSR affects the stability of upstream collusion basically hinges on the downstream competition modes. Specifically, CSR behavior facilitates (hinders) upstream collusion under downstream quantity (price) competition. For given degree of CSR and product substitutability, upstream collusion is always less stable under downstream price competition. In addition, higher degree of product substitutability obstructs collusion, regardless of downstream competition mode.
We compare collusion stability under Bertrand and Cournot duopoly with differentiated network products. Contrast to previous studies, we show that (i) the range of collusion incentive is narrower under Cournot competition than under Bertrand competition, unless network externalities are sufficiently strong; (ii) collusion in prices (quantities) is more stable than in quantities (prices) if network externalities are strong (weak); and (iii) finally, regardless of the strength of network externalities, collusion under Bertrand competition is more stable for high levels of substitutability than under Cournot competition.
New theoretical insights to tackle certain enduring and emerging issues in the context of market structure, technological innovations and exports are presented in the paper. We deal with the underlying mechanisms and outlines of theoretical frameworks in terms of a survey paper. Apart from the general survey of the literature, we specially bring to the fore the role played by technology and competition in innovation and export decisions. We also consider the issue of finance and credit constraints as determining factors behind export behaviour of firms. The results in general substantially challenge the conventional wisdom.
In an infinite-horizon oligopoly game, if moves are not synchronised across firms, sustainability of tacitly collusive equilibria becomes substantially distinct from that in a classical simultaneous-move supergame. In linear duopoly, the critical discount factor becomes higher in Bertrand than in Cournot, yet for a low discount factor with which collusion is unsustainable the equilibrium discounted streams of non-collusive profits are higher in a Bertrand game than in a Cournot game. These features are contrary to the well-known observation in synchronous-move supergames, reflecting the tendency that once a deviation path is initiated, then an asynchronous-move Cournot game converges to a competitive outcome more quickly than an asynchronous-move Bertrand game.
This paper analyses the investment incentives in a vertical separation structure with symmetric costs and its impact on firms'behaviors on the downstream market which are competing under Bertrand duopoly and are differenciated horizontally à la Hotteling. We find that when firm less value the investment made upstream, it will be more encouraged to deviate from the collusive agreement. It is also shown that firms will have more incentives to collude when the difference between their ability to use infrastructure investment will increase too.
This paper revisits the conventional doctrine that “it is easier to collude among equals”, applied in the context of vertically related markets. In a differentiated duopoly model, we study how cost-based access price regulation may hinder the sustainability of tacit collusion.
I examine the question whether cooperation in R&D among firms producing similar products leads to product market collusion. Suppose that firms engage in a stochastic R&D race while maintaining the collusive equilibrium in a repeated-game framework. Innovation under non-cooperative R&D leads to an inter-firm asymmetry, destabilizing collusion in pre-discovery and post-discovery periods. Innovation sharing under cooperative R&D preserves the symmetry and also increases total profit, thereby facilitating collusion. However, welfare may increase with cooperative R&D. I also examine the condition for collusion under licensing and compare the results. Copyright 2009 The Authors. Journal compilation 2009 Blackwell Publishing Ltd. and the Editorial Board of The Journal of Industrial Economics.
In the last decade, the analytical progress achieved in the New Keynesian literature has been remarkable. Many of the early assumptions have been relaxed, leading to medium-scale macroeconomic models that are now able to capture many features of real-world data. Nevertheless, modern-day New Keynesian models still assume, as did their early counterparts, that firms compete in the market with no tools other than their relative prices. In particular, this literature has so far neglected the consequences of extending competition between firms to the non-price dimension. This paper tries to fill this gap by enriching the canonical New Keynesian framework to include both price and non-price competition. This has important consequences for the analysis of inflation dynamics, modifying in particular the inflation-marginal cost relationship. As a general result, we show that any activity by firms that boosts demand for their products, without directly affecting their prices, dampens the overall degree of real rigidities in price-setting.
In this paper we provide a sufficient condition for collusive outcomes in a single-shot game of simultaneous price choice
in a homogeneous product market with symmetric firms and strictly convex costs. We also prove the counterintuitive result:
if the second derivative of the cost function is nonincreasing in output, it is easier to sustain collusion when the number
of firms increases.
In an oligopoly supergame, firms’ actions in prices and quantities are subject to non-negativity constraints. These constraints can obstruct the practicability of some of the tacitly collusive subgame perfect equilibria, such as single-period optimal punishment which relies indispensably upon firms’ ability to charge prices strictly below marginal costs, i.e. temporarily loss making pricing. Thereby under the presence of positive price constraints, marginal costs can serve as a ‘fudge’ to materialise such penal pricing. In this paper we briefly shed light on the effects of profit–cost ratios (or mark-ups) on the sustainability of tacit collusion.
In a repeated oligopoly with differentiated products and firms following trigger strategies, quantity competition leads to a less efficient outcome than price competition. This result contrasts with the repeated duopoly result, where price competition is less efficient under certain conditions.
Stability of collusion in differentiated oligopolies is studied without symmetry restrictions on the available strategies. It is demonstrated that if the number of firms is sufficiently large, two-phase stick-and-carrot punishment schemes apply at the highest possible discount rate with respect to collusion on the joint profit-maximizing output. If stick-and-carrot punishment schemes are used, collusive stability of the joint profit-maximizing output improves monotonically with the degree of product differentiation. The conclusions contrast with those obtained by Wernerfelt [Economics Lett. 29 (1989) 303].
This paper employs a supergame-theoretic model of collusion to analyze the effects of different levels of product differentiation on cartel stability. In contrast to earlier work, the ability of the parties to reach an agreement is assumed. Here the focus is on how greater product homogeneity increases both the gain to cheating on a collusive agreement and the magnitude of the punishments that follow a defection. The total effect on the likelihood a cartel is stable is therefore unclear. Two particular differentiated products models are then used to demonstrate that, contrary to the conventional view, greater homogeneity can reduce cartel stability.
When goods are complements or very close substitutes, more tacit collusion is supported by trigger strategies in price setting supergames than in quantity setting supergames. For moderate or poor substitutes, the situation is reversed. This contrasts with the static situation, where price setting strategies always lead to higher welfare.
The paper analyses the existence of pure strategy Nash equilibrium in price competition (or Bertrand equilibrium) in a homogeneous product market when costs are strictly convex and proves that if output is demand determined such equilibrium always exists. This paper also characterises such equilibria and shows that if firms are identical such equilibria are necessarily non-unique. However for firms with asymmetric costs it can be unique or non-unique.
This paper examines the effect of increased product substitutability on quantity-setting firms' ability to sustain tacit collusion in a market. It uses a general demand function and the trigger strategy of Friedman (Friedman JW. 1971. A non-cooperative equilibrium for supergames. Review of Economic Studies 38: 1-12) to show that while increased product substitutability hinders sustainability of tacit collusion in the case of linear and concave demand functions, it may either hinder or facilitate firms' ability to sustain tacit collusion in the case of convex demand functions. Thus, this paper adds to the growing view that one must use a case-by-case analysis in judging whether firms in more homogenous product markets find it easier or harder to tacitly collude. Copyright © 1999 John Wiley & Sons, Ltd.
Corruption in the public sector erodes tax compliance and leads to higher tax evasion. Moreover, corrupt public officials abuse their public power to extort bribes from the private agents. In both types of interaction with the public sector, the private agents are bound to face uncertainty with respect to their disposable incomes. To analyse effects of this uncertainty, a stochastic dynamic growth model with the public sector is examined. It is shown that deterministic excessive red tape and corruption deteriorate the growth potential through income redistribution and public sector inefficiencies. Most importantly, it is demonstrated that the increase in corruption via higher uncertainty exerts adverse effects on capital accumulation, thus leading to lower growth rates.
In a letter previously published in this journal, Rothschild (1992)[Rothschild, R., 1992. On the sustainability of collusion in differentiated duopolies. Economics Letters 40, 33–37] claims to obtain results in contrast with those obtained by Deneckere (1983)[Deneckere, R., 1983. Duopoly supergames with product differentiation. Economics Letters 11, 37–42]. We show however that, taking correctly into account non-negativity constraints on output, Rothschild's formulation yields the same conclusions as Deneckere's.
We consider a rule designed to maintain collusion in a duopoly made up of firms producing differentiated products. We analyse both price-setting and quantity-setting models, and show that product differentiation plays a crucial role in determining the effectiveness of this rule. Our results contrast with those obtained by earlier writers.
In a simple oligopoly with quantity setting firms, we analyze the conditions under which more product differentiation makes tacit collusion easier. It is found that the net effect can go either way.
Presents a non-cooperative equilibrium concept, applicable to supergames, which fits John Nash's non-cooperative equilibrium and also has some features resembling the Nash cooperative solution. Description of an ordinary game; Definition and discussion of a non-cooperative equilibrium for supergames; Description of supergame and supergame strategies; Information on the Cournot strategy.
The stability of collusion is analysed for a family of demand functions whose curvature is determined by a parameter varying between zero and infinity. When the number of firms is low, firms may prefer to act as quantity setters in order to increase cartel stability if demand is sufficiently convex. Otherwise, price-setting behaviour enhances their ability to collude. As the number of firms tends to infinity, Cournot behaviour is preferable to Bertrand behaviour in order to stabilize collusion, independently of the characteristics of market demand. Copyright 1996 by Blackwell Publishing Ltd and the Board of Trustees of the Bulletin of Economic Research
- R Deneckere
Deneckere, R. (1984). 'Corrigenda', Economics Letters, 15, pp. 385–7.