On the Optimal Number of Firms in the Commons: Cournot vs Bertrand

Dragone, Davide; Lambertini, Luca; Palestini, Arsen; Tampieri, Alessandro Working Paper On the Optimal Number of Firms in the Commons: Cournot vs Bertrand Quaderni - Working Paper DSE, No. 856 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Dragone, Davide; Lambertini, Luca; Palestini, Arsen; Tampieri, Alessandro (2012) : On the Optimal Number of Firms in the Commons: Cournot vs Bertrand, Quaderni - Working Paper DSE, No. 856, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/3908 This Version is available at: https://hdl.handle.net/10419/159695 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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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/ On the Optimal Number of Firms in the Commons: Cournot vs Bertrand Davide Dragone Luca Lambertini Arsen Palestini Alessandro Tampieri Quaderni - Working Paper DSE N° 856 On the Optimal Number of Firms in the Commons: Cournot vs Bertrand Davide Dragone#, Luca Lambertini#, Arsen Palestinixand Alessandro Tampieri# # Department of Economics, University of Bologna Strada Maggiore 45, 40125 Bologna, Italy [email protected]; [email protected]; [email protected] § MEMOTEF, University of Rome La Sapienza via del Castro Laurenziano 9, I-00161 Roma, Italy [email protected] December 11, 2012 Abstract We revisit the debate on the optimal number o …rms in the commons in a di¤erential oligopoly game in which …rms are either quantityor price-setting agents. Production exploits a natural resource and involves a negative externality. We calculate the number of …rms maximising industry pro…ts, …nding that it is larger in the Cournot case. While industry structure is always ine¤cient under Bertrand behaviour, it may or may not be so under Cournot behaviour, depending on parameter values. The comparison of private industry optima reveals that the Cournot steady state welfare level exceeds the corresponding Bertrand magnitude if the weight of the stock of pollution is large enough. Keywords: natural resources, oligopoly, pollution, tragedy of commons JEL Codes: C73, L13, Q20, Q51 We thank the audience at IFAC CAO’12 (Rimini) for precious comments. The usual disclaimer applies. 1 1 Introduction The usual approach to the economics of the environment treats externalities and the extraction of natural resources separately, and in the latter case compares open access (or equivalently, perfect competition) against monopoly.1 Here we propose a uni…ed approach to the two aspects of the industrial exploitation of the environmen, using a homogeneous good oligopoly in which …rms may set either quantities or prices to maximise pro…ts, and their productive activities require the use of a renewable resource and emit pollutants. Each of these two aspects has indeed received attention in the literature, either in static or in dynamic oligopoly models,2but, to the best of our knowledge, the joint analysis of resource extraction and pollution has not. Our analysis will abstract from the possibility of regulating …rms’interaction via Pigouvian taxation/subsidization3and/or pollution rights,4to focus on the issue of the optimal number of …rms in the commons. This problem lies at the intersection between the well known discussion about the tragedy of the commons (Gordon, 1954; Hardin, 1968) and the standard approach to the entry process belonging to the theory of industrial organization (Novshek, 1980; Suzumura and Kiyono, 1987; Mankiw and Whinston, 1990). The backbone of this discussion is the fact that, while in absence of any external e¤ects increasing competition (and therefore industry output) increases welfare, if industrial activities exploit natural resources and/or imply the emission of pollutants then the socially optimal degree of concentration of such an industry is determined by the balance between the price e¤ect and the environmental one (Cornes and Sandler, 1983; Cornes, Mason and Sandler, 1986; Karp, 1992; Mason and Polasky, 1997). We revisit this issue in a di¤erential game in which we assess the privately optimal structure (maximising industry pro…ts) against the socially optimal industry structure (maximising social welfare), given the pro…t-maximising behaviour of …rms, under both Cournot and Bertrand competition. Industry structure is always socially ine¢ cient under Bertrand behaviour, while it may or may not be so under Cournot behaviour, depending on the environment’s e¢ ciency in ab1See Dasgupta and Heal (1979), Kemp and Long (1980), Pearce and Turner (1989), Tisdell (2009) and Anderson (2010), inter alia. 2The related literature is too large to be cited entirely. See, inter alia, McMillan and Sinn (1984), Sinn (1984), Katsoulacos and Xepapadeas (1995) and Fujiwara (2009). 3See Karp and Livernois (1992) and Benchekroun and Long (1998, 2002). 4See von der Fehr (1993) and Sunnevåg (2003). 2 sorbing pollution. Then, we establish that (i) the privately optimal structure in Cournot exceeds its counterpart in Bertrand, and consequently (ii) social welfare in the private optimum can be higher at the Cournot equilibrium, if the weight of pollution in the social welfare function is su¢ ciently high. The basic model is laid out in section 2. The non-cooperative equilibrium between pro…t-maximizing …rms is outlined in section 3. Section 4 contains the analysis of the social planning equilibrium. The two regimes are comparatively assessed in section 5, while the mixed setting is investigated in section 6. Concluding remarks are in section 7. 2 The setup Consider an oligopoly market over an in…nite (continuous) time horizon, t2[0;1);in which n2…rms supply a homogeneous good, whose market demand function is p(t) = aQ(t)(1) at any time t2[0;1);with a > 0being a positive constant parameter measuring the reservation price and Q(t) = Pn i=1 qi(t)being the sum of all …rms’output levels. Production takes place at decreasing returns to scale, with the same technology being common to all …rms alike, so that …rm i’s instantaneous cost function is Ci(t) = cq2 i(t);with the constant c > 0.5 The production of the …nal output goes along with a negative environmental externality whose instantaneous level is  (t) = S2(t)=2;with  > 0and S(t)evolving over time according to the following dynamics: dS (t) dt   S(t) = bQ (t)S (t)(2) where  > 0is the decay rate of the stock and bis a positive constant. The Instantaneous consumer surplus CS (t)is measured by the area below the demand function and above market price p(t);minus the externality  (t): CS (t) = Q2(t) 2S2(t) 2:(3) 5We could have speci…ed the cost function as Ci(t) = zqi(t) + cq2 i(t);with z > 0:This would be a useless complication, however, as one could as well think of the vertical intercept of the demand function as a=baz; whereby the ensuing analysis would reproduce unmodi…ed. 3 It is worth noting that a contraction of output has ambiguous consequences over consumer surplus, due to the presence of a negative externality proportional to the output: on the one hand, shrinking output goes along with increasing market price, which is harmful; on the other hand, it entails reducing the environmental externality, which is desirable. The balance between these components will play a key role in the remainder of the analysis. Additionally, the production of the …nal good makes use of a renewable natural resource whose stock X(t)follows the state equation: dX (t) dt   X(t) = X (t)vQ (t);(4) with constants and vstrictly positive. The instantaneous social welfare function, de…ned as the sum of industry pro…ts and consumer surplus, writes as follows: SW (t) = n X i=1 i(t) + Q2(t) 2(5) S2(t) 2+X(t) where i(t) = [p(t)c]qi(t)is …rm i’s instantaneous pro…t function. In the remainder of the paper, we investigate the non-cooperative unregulated open-loop game where …rms compete either à la Cournot-Nash or à la Bertrand, alternatively, to maximise individual pro…ts. In both cases, …rm ichooses its strategy (either quantity or price) to maximise the discounted individual pro…t ‡ow: Ji(t) = Z1 0 i(t)etdt (6) s.t. the state equations (2) and (4), and the initial conditions S(0) = S0and X(0) = X0:Parameter  > 0represents the constant discount rate common to all …rms in the industry. 3 The Cournot-Nash game Here we characterise the open-loop equilibrium of the …rst game, where all …rms are private and compete à la Cournot-Nash to maximise individual pro…ts. Our …rst objective is to prove the following claim: 4 Proposition 1. The game among pro…t-maximising …rms is a linear state one, and therefore its open-loop Cournot-Nash solution is strongly time consistent. Proof. The current value Hamiltonian of …rm iis: Hi(t) = i(t) + i(t) S(t) + i(t) X(t)(7) where i(t)and i(t)are the co-state variables associated with the dynamics of pollution and the natural resource, respectively. The following system illustrates the set of …rst order conditions on controls and the associated co-state equations (omitting henceforth the time argument for brevity): @Hi @qi =a2 (1 + c)qiQi+bivi= 0 (8) @Hi @S =  ii,  i= (+)i(9) @Hi @X = ii, i= (+)i(10) where QiPj6=iqjis the amount of instantaneous output collectively supplied by all rivals of …rm iat any given time. Clearly, (8-10) jointly imply that the optimal output of …rm inever depends on the states. The intuitive reason is that …rms - being unregulated pro…t maximising entities - are completely uninterested in the amount of pollution and the stock of the resource and consequently behave as if the two-sided tragedy of commons did not exist. From a strictly technical standpoint, one can easily check that @2Hi @qi@S =@2Hi @S2= 0 (11) as well as @2Hi @qi@X =@2Hi @X2= 0 (12) and therefore the game is indeed a linear state one (cf. Dockner et al., 2000, p. 188), yielding a subgame perfect or strongly time consistent Nash equilibrium under the open-loop information structure. 5 Accordingly, from (10) one obtains i= 0 for all i= 1;2;3; :::N at any time during the game. Then, from (8), one …nds i=2 (1 + c)qi+Qia b:(13) Then, di¤erentiating w.r.t. time, imposing symmetry across quantities (qj= qi=qfor all i; j) and using (9), the control equation obtains:  q=(+) [q(n+ 1 + 2c)a] 2c+n+ 1 :(14) Imposing stationarity, we have qCN =a= (n+ 1 + 2c);which coincides with the solution of the static game. Superscript CN stands for Cournot-Nash. Of course the same solution obtains immediately by observing that the system of co-state equations (9-10) admits the solution i=i= 0 for all i= 1;2;3; :::N at all times, whereby the …rst order condition (8) indeed delivers qCN =a= (n+ 1 + 2c)throughout the game.6 Before proceeding any further, we brie‡y evaluate the stability properties of the dynamic system (2-4-14), by looking at the associated Jacobian matrix: J= 2 6 6 6 6 6 6 6 6 4 @ S @S @ S @X @ S @q @ X @S @ X @X @ X @q @ q @S @ q @X @ q @q 3 7 7 7 7 7 7 7 7 5 (15) that is, J=2 40bN 0vN 0 0 + 3 5(16) whose eigenvalues are 1= < 0; 2= > 0; 3=+ > 0:(17) 6This also implies that i= 0 for all ithroughout the game, as is easily veri…ed from (13). Thus, the transversality conditions lim t!1 iS= lim t!1 iX= 0 are trivially satis…ed for all i: 6 Accordingly, we can state: Proposition 2. The Cournot-Nash equilibrium of the open-loop game is a saddle point. The corresponding amount of pollution and the residual volume of natural resource obtain, respectively, from  S= 0 and  X= 0:7 SCN =nab (2c+n+ 1) ;(18) XCN =nav (2c+n+ 1) :(19) From the above expressions we can draw: Lemma 3. Since lim n!1 SCN =ab ; lim n!1 XCN =av ; open access implies positive and …nite volumes of resource and pollution at the steady state. In particular, the second of the above limits reveals that open access does not lead to resource extinction. The per-…rm pro…ts and social welfare in steady state are CN =a2(1 + c) (2c+n+ 1)2;(20) SWCN =na 2 (2c+n+ 1)22(21) with a(n+ 2 (1 + c)) 2nb2 +2v(2c+n+ 1) 2:(22) The above expression reveals the following result. Lemma 4. The condition  > brn n+ 2 (1 + c)CN SW 7Then, one can also easily show that the feedback equilibrium based upon the linear value function Vi(S; X) = !1+!2S+!3Xand the corresponding Bellman equation Vi(S; X) = maxqi[i+@Vi=@S dS=dt +@Vi=@X dS=dt]is indeed SCN ; XCN ; qCN : 7 scale. Exactly the opposite would intuitively apply if the marginal cost were constant, as in such a case the Bertrand-Nash equilibrium would coincide with perfect competition and consequently industry output would be higher than the Cournot-Nash one, with obvious consequences on industry pro…ts. However, by the same token, consumer surplus and the steady state volume of natural resource are both higher under Cournot competition than under Bertrand competition. A di¤erent exercise can be envisaged to compare the two industry output given their respective optimal industry structures nCN and nBN ;to evaluate the di¤erence Q=QBN nBN QCN nCN  =(52) + 2cp(+ 8c) 4 (c); whose numerator is positive for all c >  (and conversely), so that Q < 0 everywhere.10 For the aforementioned reasons, this has in principle ambiguous consequences on welfare. Hence we must evaluate SW WBN nBN WCN nCN  :(53) In the special case = 0 (i.e., under average cost pricing under Bertrand behaviour), we have: SW / a 2b2+ 2v2(1 + 2c)2cv2<0(54) for any  > bp: For any 2(0;4=3] ;we have instead:11 SW = (a; b; c; v; ; ; ) (a; b; c; v; ; ; )(55) with  (); ()>0;so that SW > 0for all  >  ()= (). Accordingly, we may state: Proposition 8. In correspondence of the optimal industry structure nKN ; K=B; N, the steady state social welfare level is higher under Cournot behaviour for all admissible levels of ; if average cost pricing prevails in the Bertrand game. Otherwise, for positive values of ; steady state social welfare is higher under Cournot competition if is high enough. 10 The possibility for Cournot to accomodate more …rms (and therefore deliver a higher output) than Bertrand has been highlighted in a static game by Cellini, Lambertini and Ottaviano (2004). 11 We omit the expressions  ()and  ()for brevity. These can be reconstructed from the de…nition of SW in (53). 14 7 Conclusions We have analysed a di¤erential oligopoly game in which environmental externalities and the exploitation of natural resources combine in a single framework. Considering prices or quantities alternatively as the …rms’strategic instruments, we have assessed the privately optimal number of …rms against the socially optimal one, showing the emergence of an ambiguous conclusion in the Cournot setup. Conversely, under Bertrand behaviour the privately optimal degree of concentration is de…nitely too large from the social standpoint. Relatedly, taking as a benchmark the privately optimal industry structure, we have shown that the relative size of welfare levels at the steady states of the two models depends on the capability of the environment to absorb polluting emissions. The foregoing analysis has been carried out assuming any form of regulation away. The study of the interplay between environmental policy, …rms’ strategic behaviour and the (in)e¢ ciency of the resulting industry structure in the commons is left for future research. 15 REFERENCES Anderson, D. (2010). Environmental Economics and Natural Resource Management. Third Edition. 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