Cournot Vs Stackelberg Equilibria With Entrepreneurial and Labour Managed Firms

Lambertini, Luca Working Paper Cournot Vs Stackelberg Equilibria With Entrepreneurial and Labour Managed Firms Quaderni - Working Paper DSE, No. 217 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Lambertini, Luca (1995) : Cournot Vs Stackelberg Equilibria With Entrepreneurial and Labour Managed Firms, Quaderni - Working Paper DSE, No. 217, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/5108 This Version is available at: https://hdl.handle.net/10419/159060 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/3.0/ COURNOT VS STACKELBERG EQUILIBRIA WITH ENTREPRENEURIAL AND LABOUR MANAGED FIRMS Luca Lambertini Dipartimento di Scienze Economiche# Università degli Studi di Bologna Strada Maggiore 45 40125 Bologna Italy tel 39-51-6402600 fax 39-51-6402664 e-mail [email protected] and Linacre College Oxford OX1 3JA United Kingdom e-mail [email protected] Abstract The issueof equilibrium selection in a duopolygame between a profit maximizing and a labour managed firm is addressed under either price or quantity competition with product differentiation.If firms can choosethe timing of moves beforecompeting in therelevant market variable, the Bertrand game yields multiple equilibria, while the Cournot game has a unique subgame perfectequilibrium with the profit maximizing firm in the leader’s role and the labour managed firm in the follower’s role. Due to a lower total output, the Cournot-Stackelberg equilibrium yields a lower level of social welfare as compared to the simultaneous equilibrium. This reduces the incentive to transform an LM duopoly into a mixed one. JEL classification: D43, D92, L13, L20 Keywords: extended game, sequential play Acnowledgements I would like to thank Flavio Delbono, Vincenzo Denicolò, Paolo Garella and Gianpaolo Rossini for helpful comments and discussion. The usual disclaimer applies. #Please send correspondence to the Italian address 1. Introduction A large body of literature deals with the issue of choosing roles in sequential duopoly games.Inthecontextofduopolisticcompetitionbetweenprofitmaximizing(PM)firms,Gal-Or (1985)andDowrick(1986)showthat,providedfirmsaresymmetric,theslopeoftheirrespective reaction functions in the relevant strategic variable, i.e., either price or quantity, determines whether they prefer to act as a leader or a follower. Specifically, both firms would prefer to be the leader (follower) in quantity (price) setting games if reaction functions are downward (upward) sloping, due to the presence of strategic substitutability (complementarity) between goods (see Bulow et al., 1985). The results reached by the above contributions are extended to the case of differentiated products by Gal-Or (1985) and Boyer and Moreaux (1987). In arecent paper,Okuguchi (1993b) investigates the preferencesof labourmanaged (LM) firms as for the distribution of roles under both Bertrand and Cournot competition and product differentiation, finding out that, in sharp contrast to what happens when only entrepreneurial firms are involved, in the case of a pure LM duopoly, reaction functions are upward sloping regardlessof the kind of competition, be that inprices or quantities. Hence, both LM duopolists would prefer to act as a follower, independently of the strategic variable being set. Even though the comparison between the payoffs accruing to duopolists in simultaneous and sequential games, as well as the conclusions drawn from it, is relevant in itself, it does not provide any answer to the main question, namely, whether firms’ preferences would allow for any of the sequential or simultaneous equilibria to endogenously emerge as the equilibrium of the underlying game one could envisage, i.e., a game where firms are first required to announce the timing of their respective moves and then proceed to set the relevant variable in order to maximize their own objective function in the basic market game. This issue has been tackled in a very influential paper by Hamilton and Slutsky (1990). They embed simultaneous and sequential play into an extended game with observable delay where players must set both the strategicvariableofthebasicgameandthetimetosetthatvariable.Thelatter processisactually 1 a logical preplay stage which is not observed. If players decide to move at the same time, a simultaneous equilibrium is observed, and viceversa. It is noteworthy that the decision to play early rather than at a later stage is not sufficient to yield Stackelberg leadership, since an analogous decision by the rival determines the emergence of a simultaneous Nash equilibrium. Thus, a Stackelberg equilibrium (or sequential play) with one player moving first and the rival second will be the only subgame perfect equilibrium of the extended game if only one of the two possible sequential play outcomes Pareto-dominates the simultaneous play outcome (Hamilton and Slutsky, 1990, Theorem IV, p.37). Otherwise, when both players share the same preferences over the sequence of moves and the follower’s payoff dominates that associated with simultaneous play, then both sequential equilibria (as well as a mixed strategy one) are subgame perfect equilibria of the extended game, so that in principle it is impossible to know which of them will be actually observed (Hamilton and Slutsky, 1990, Theorem III, p.36). Applyingthe tools provided by Hamilton and Slustky (1990), I want to address a question which so far, to the best of my knowledge, has remained neglected, i.e., which preferences characterize a mixed duopoly game between a profit maximizing and a labour managed firm, and consequently which kind of equilibrium one can expect to obtain in such a game if firms can decide the timing of moves before proceeding to compete in prices or quantities. The behaviour of LM firms in mixed oligopolies has been described by several authors (see, inter alia, Cremer and Crémer, 1992; Delbono and Rossini, 1992; Rossini and Scarpa, 1993; Okuguchi, 1993a). They have highlighted the peculiar behaviour of LM firms under quantity competition, yielding an upward sloping reaction function1instead of the usual downward sloping one characterizing the PM firm. Nevertheless, all these contributions investigate to various aims simultaneous play under either quantity or price setting behaviour. Iwill showthat, whena preplaystage inthe sense ofHamilton andSlutsky (1990) isintroduced, 1. However, the reaction function of an LM firm is not necessarily upward sloping. See Miyamoto (1982, p.13). 2 (i) simultaneous play is not to be expected under neither form of competition; (ii) Cournot behaviouryieldsastheuniquesubgameperfectequilibriumoftheextendedgametheStackelberg equilibrium with the PM firm moving first, and (iii) Bertrand behaviour leads to multiple equilibria in which both firms would prefer to move late or play in mixed strategies. These results have some interesting implications as for the issue of reforming Eastern Europeaneconomies.Delbono andRossini (1992)evaluatethe feasibilityof alternativereforms of LM markets consisting in the passage to a mixed oligopoly or a horizontal merger where the resulting firm maximizes an objective function in which a positive weight is assigned to either entrepreneurial profit or social welfare. In analysing the case of a mixed duopoly, they only consider simultaneous Nash equilibria. In the present paper, it is shown that only Stackelberg equilibria should be taken into account. Hence, it turns out that a reform based on either the privatization or the nationalization of a labour managed firm implies a smaller social gain than it could be expected on the basis of previous literature. The remainder of the paper is structured as follows. Betrand competition is described in Section 2. Section 3 is devoted to Cournot competition. Policy implications are discussed in Section 4. Finally, Section 5 contains concluding comments. 2. Bertrand competition In order to safeguard the comparability of what follows with at least a part of the existing literature, I basically adopt the same symbology and assumptions as in Okuguchi (1993b). The magnitudes related to the PM and LM firms are identified as Pand C, respectively. Both firms produce through the following technology: where liis the amount of labour employed by firm iand xiis the quantity produced by the same li=hi(xi), i=C,P(1) 3 firm. The technology is fully characterized by the following derivatives: i.e.,themarginalproductivityoflabourisdecreasing.Firmsoperateinamarketfordifferentiated goods, whose demand is where (see Okuguchi, 1993b, pp.2-3):2 The inequalities in (4.1) state that (i) an increase in firm i’s price induces a decrease in the demand for her own product, (ii) the two goods are substitutes, and (iii) the own price effect is larger than the cross price effect. The inequalities in (4.2) are needed for the reaction function of the LM firm to be positively sloped. Since under the above assumptions Okuguchi (1993a,b) has shown that in a Bertrand hi’>0,hi">0,(2) xi=gi(pi,pj), i,j=C,P,i≠j,(3) ∂gi/∂pi≡gii<0,∂gi/∂pj≡gji>0,−gii>gji;(4.1) ∂2gi ∂pi∂pj≡gij i≤0,gji+pigij i>0. (4.2) 2. These assumptions, as well as those introduced in the remainder of the paper, hold for instance when linear demand functions are considered. 4 setting the reaction function of an LM firm is positively sloped irrespectively of the nature of the rival, I can confine myself to investigate the characteristics of the entrepreneurial firm’s reaction function. I am going to prove the following: LEMMA 1. Under Bertrand competition, the reaction function of the profit maximizing firm is upward sloping. PROOF. The objective function of the PM firm is the following: where kPdefines the entrepreneurial firm’s fixed cost. The first order condition for profit maximization w.r.t. price is: Assume the second order condition is satisfied. It is known (see Bulow et al., 1985) that the slope of the reaction function has the same sign as the derivative of (6) w.r.t pC: Accordingly, it is sufficient to determine the sign of πP B=pPgP(pC,pP)−hP(xP)−kP(5) ∂πP B ∂pP=gP(pC,pP)+pPgP P−hP ’gP P=0. (6) sign ∂pP ∂pC=sign ∂2πP B ∂pP∂pC(7) 5 on the basis of the above assumptions, it is quickly established that the sign of (8) is positive. Hence, the reaction function of the PM firm in the price space is upward sloping. Q.E.D. Provided that the reaction function of the PM firm is positively sloped, as claimed in Lemma 1, and the reaction function of the LM firm is also increasing, as shown by Okuguchi (1993b), I am going to show what is stated in the following: PROPOSITION1.TheextendedBertrandgamebetweenaprofitmaximizingfirmandalabour managed firm has multiple equilibria. None of them is simultaneous. PROOF.Since both reaction functions are positively sloped, this setting is a special case of the general situation depicted by Hamilton and Slutsky (1990, pp.36-41) in their Theorems III, V(Aii) and VI. According to these theorems, when both reaction functions are increasing the extended game with observable delay, where players first choose the timing of moves and then proceed to play, has multiple equilibria. Namely, both sequential play are subgame perfect equilibria; moreover, there exists a mixed strategy equilibrium in which firms randomize over the strategies "moving first" and "moving second". This is due to the fact that both reaction functions intersect the Pareto superior set, i.e., the set of all pair of prices yielding payoffs that dominate those associated with the simultaneous equilibrium. Q.E.D. ∂2πP B ∂pP∂pC=gC P+pPgPC P−hP ’gPC P−hP "gP PgC P;(8) 6 3. Cournot competition In this Section, optimization w.r.t. quantity is analised. If the domain of the demand function (3) is a rectangular region, it can be inverted to obtain: with Assumption (10.1) is borrowed from Okuguchi (1993b, p.4). Assumption (10.2), which is a bit tighter than the corresponding condition in Okuguchi (1993b, p.4), and is borrowed from Okuguchi (1993a, p.29), implies that firm i’s marginal revenue decreases as her rival’s output increases. Provided firm iacts as a profit maximizer, this condition also implies that her own reaction function is negatively sloped (see Novshek, 1985; Dixit, 1986; Okuguchi, 1993a, inter alia).3ProvidedthatthereactionfunctionoftheLMfirmisupwardsloping(Okuguchi,1993a,b), the following holds: PROPOSITION 2. The Stackelberg equilibrium with the profit maximizing firm moving first and the labour managed firm moving second is the only subgame perfect equilibrium of the extended Cournot game. pi=fi(xi,xj), i,j=C,P,i≠j,(9) ∂fi/∂xj≡fji<0; (10.1) ∂2fi ∂xi∂xj≡fij i∈[0,−fji xi[. (10.2) 3.Cournotbehaviourmayinduceaprofitmaximizingfirmtoconsiderherrivalsasstrategic complements.Thishappenswhenalargedominantfirmcompetesagainstapopulationofsmaller rivals. See Bulow et al. (1985, p.500). 7 Appendix A.1. The leader’s output in the LM duopoly The solution to the leader’s problem in the pure LM duopoly is given by: where The output of the follower, firm j, can be obtained through her reaction function (17). Since it must be that the following constraint is to be satisfied: A.2. The leader’s output in the mixed duopoly When the PM firm plays the leader’s role in the mixed duopoly game, her production amounts to: where xi=2 3a−2k 3a+(η+φ) 1/3−a(6k+3a2−(2F/a−2a)2)/3 [27a3(η + φ)]1/3(a.1) η=−8k3+42a2k2−6a4k−a6 27a3φ= 1 3a√ 96a2k3−20k4−24a4k2+2a6k 3(a.2) a>xi+xj, a2>k(3+2√2). (a.3) xP=a 2    3 2+1 ψ1/3+ψ1/3 4   ,(a.4) 14 Again,thequantityproducedbythefollower,inthiscasetheLMfirm,canbecomputedresorting to her own reaction function (20). Finally, the condition that must be met in order for market price to be positive at equilibrium is the following: ψ=−(16k+a2) 8+a√ 8k2+a2k 2.(a.5) a2>2k.(a.6) 15 References Boyer, Marcel and Michel Moreaux, "On Stackelberg Equilibria with Differentiated Products: The Critical Role of the Strategy Space", Journal of Industrial Economics, XXXVI, 217-30, 1987. Bulow, Jeremy, John Geanakoplos and Paul Klemperer, "Multimarket Oligopoly: Strategic Substitutes and Complements", Journal of Political Economy,93, 488-511, 1987. Cremer, Helmut, and Jacques Crémer, "Duopoly with Employee-Controlled and Profit-Maximizing Firms: Bertrand vs Cournot Competition", Journal of Comparative Economics,16, 241-58, 1992. Delbono,Flavio,andGianpaoloRossini,"CompetitionPolicyvsHorizontalMergerwithPublic, EntrepreneurialandCooperativeFirms",JournalofComparativeEconomics,16,226-40, 1992. Delbono, Flavio, and Carlo Scarpa, "Upward Sloping Reaction Functions under Quantity Competition in Mixed Oligopolies", Dipartimento di Scienze Economiche, Bologna, mimeo; to appear in Bulletin of Economic Research, 1995. Dixit, Avinash, "Comparative Statics for Oligopoly", International Economic Review,27, 103-22, 1986. Dowrick, Steve, "von Stackelberg and Cournot Duopoly: Choosing Roles", Rand Journal of Economics,17, 251-60, 1986. Gal-Or,Esther, "First Mover and Second Mover Advantages", International Economic Review, 26, 649-53, 1985. 16 Hamilton, Jonathan H., and Steven M. Slutsky, "Endogenous Timing in Duopoly Games: Stackelberg or Cournot Equilibria", Games and Economic Behavior,2, 29-46, 1990. Kahana, Nava, "Do Multiplant Labor-Managed Monopolies Exist?", Journal of Comparative Economics, 18, 198-201, 1994. Miyamoto, Yoshinari, "A Labour-Managed Firm’s Reaction Function Reconsidered", Economic Research Paper n. 218, University of Warwick, Coventry, 1982. Novshek, William, "On the Existence of Cournot Equilibrium", Review of Economic Studies, LII, 85-98, 1985. Okuguchi,Koji,"CournotOligopolywithProfit-Maximizing andLabor-ManagedFirms",Keio Economic Studies,30 (1), 27-38, 1993a. Okuguchi, Koji, "Follower’s Advantage in Labor-Managed Duopoly", Keio Economic Studies, 30 (2), 1-5, 1993b. Rossini, Gianpaolo, and Carlo Scarpa, "The Transformation of a Labour Managed Monopoly: Competition Policy vs Mixed Firms", Journal of International and Comparative Economics,2, 191-206, 1993. Singh, Nirvikar, and Xavier Vives, "Price and Quantity Competition in a Differentiated Duopoly", Rand Journal of Economics,14, 546-54, 1984. 17