Exploration for Nonrenewable Resources in a Dynamic Oligopoly: An Arrovian Result

Lambertini, Luca Working Paper Exploration for Nonrenewable Resources in a Dynamic Oligopoly: An Arrovian Result Quaderni - Working Paper DSE, No. 859 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Lambertini, Luca (2013) : Exploration for Nonrenewable Resources in a Dynamic Oligopoly: An Arrovian Result, Quaderni - Working Paper DSE, No. 859, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/3683 This Version is available at: https://hdl.handle.net/10419/159698 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. 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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/ Exploration for Nonrenewable Resources in a Dynamic Oligopoly: An Arrovian Result Luca Lambertini Quaderni - Working Paper DSE N° 859 Exploration for Nonrenewable Resources in a Dynamic Oligopoly: An Arrovian Result Luca Lambertini Department of Economics, University of Bologna Strada Maggiore 45, 40125 Bologna, Italy [email protected] January 6, 2013 Abstract I investigate two versions of a differential Cournot oligopoly game with nonrenewable resource exploitation, in which each firm may either exploit its own private pool or exploit a common pool jointly with the rivals. Firms use a deterministic technology to invest in exploration activities. In both models, there emerges that (i) the individual exploration effort is higher when each firms has exclusive rights on a pool of its own, and (ii) depending on the assumptions on technology and demand, the aggregate exploration effort is either constant or increasing in the number of firms. JEL Codes: C73, L13, Q30 Keywords: differential games, natural resources, oligopoly 1 1 Introduction The point of departure of the literature on natural resource economics is that unregulated profit-seeking firms will not, in general, spontaneously internalise the consequences of their behaviour on the environment, much the same as we know about polluting emissions. This raises the issue of extinction/exhaustion. Indeed, a large part of the debate on nonrenewables includes exploration into the picture, with and without uncertainty affecting the exploration process (see Peterson, 1978; Deshmuk and Pliska, 1980; Arrow and Chang, 1982; Mohr, 1988; and Quyen, 1988, 1991). In the early literature, exploration is motivated by two reasons. The first is the incentive to obtain information about the uncertain size or features (e.g., quality) of the resource stock; the second is the incentive to abate extraction costs through exploration, in situations where the level of such costs is inversely related with the size of proven reserves. In Mohr (1988), where uncertainty is assumed away and the extraction cost is nil, the investments in expensive exploration activities are driven by an incentive to preempt rivals by appropriating some portion of a still unexplored common pool resource, thereby ensuring exclusive property rights on this portion and preventing its use by other firms. 1 A recent contribution (Boyce and Vojtassak, 2008), adds costly exploration under uncertainty into the framework known as the theory of ‘oil’igopoly dating back to Salant (1976), and further developed by Loury (1986) and Polasky (1992, 1996). This approach investigates the relationship between known reserves, production and exploration incentives. It predicts that firms holding larger proved reserves will tend to produce outputs which are larger in absolute size but smaller as a proportion of their reserves as compared to rivals with smaller proved reserves. Adding exploration activities to the model, Boyce and Vojtassak (2008) conclude that firms whose proved reserves gets exhausted before their owners are able to convert their unproved reserves into proved ones have a strict incentive to overinvest in exploration activities, a feature which should be easily observed, and this empirical prediction is consistent with available data concerning the last decades. Another approach deals with the search for the so-called backstop technologies, i.e., substitute technologies using renewable (and possibly, but not necessarily, green) resources replacing the traditional ones. The incentives 1 For an overview of the literature, see Lambertini (2013). 2 of firms or countries to devote R&D efforts in search substitutes of an exhaustible resources have also been investigated by a number of authors (Davidson, 1978; Hoel, 1978; Dasgupta, Gilbert and Stiglitz, 1983; Olsen, 1988; Harris and Vickers, 1995). Under perfect certainty (i.e., if the nature of the innovation is known a priori) and the date of its discovery is deterministic), importers may strategically affect the extraction path of the exporters by manipulating the timing of innovation. 2 My aim here is to set up two different models describing differential games taking place in a Cournot industry, in which firms exploit a nonrenewable resource and may invest in costly exploration activities to enlarge the stock of resource. This is modelled under two alternative assumptions, either allowing each firm exclusive rights on the exploitation of a private pool, or compelling all firms to extract the resource from a common pool. In both versions of the model, and under both assumptions, there emerge a multiplicity of steady state equilibria, among which one is the ‘myopic’ outcome leading to depletion, while another shows the existence of spontaneous incentives to invest in exploration activities even in absence of a dedicated regulatory policy. In this respect, the present analysis offer results largely differing from those typically emerging in dynamic market models dealing with polluting emissions (see, e.g., Benchekroun and Long, 1998, 2002). Comparing the firms’ efforts in the two alternative scenarios where either private pools are exploited or a single pool exists, there emerges - not surprisingly - that the public nature of the common pool curtails firms’ incentives to invest in exploration as compared to the situation in which the benefits of private access can be fully appropriated by each firm, a classical free-riding phenomenon consistently characterising the issue of privately financing the supply of a public good. 3 An additional implication of the ensuing analysis is close in spirit to one of the backbones of the literature on R&D incentives, related with the consequences of market structure (or the intensity of competition) and the innovation incentives at the industry level, dating back to the debate between Schumpeter (1942) and Arrow (1962). In this respects, the two models investigated here convey the message that aggregate exploration efforts are non-decreasing in the number of firms operating in the industry. Although 2 On this, see, in particular, the debate between Dasgupta, Gilbert and Stiglitz (1983), Gallini, Lewis and Ware (1983) and Olsen (1988). 3 See Fershtman and Nitzan (1991), Xie (1997), Karp and Lee (2003) and Cellini and Lambertini (2007), among others. 3 preliminary, as the settings proposed here are far from general, this conclusion clearly speaks in favour of the Arrovian position. 2 Model I Examine an industry consisting of nsingle-product firms exploiting a nonrenewable resource over continuous time t∈[0,∞)to produce a final good which is differentiated as in Singh and Vives (1984), so that each firm ifaces the instantaneous demand function p i (t) = a−q i (t)−sQ −i (t),(1) where Q −i (t) =  j=i q j (t)is the collective output of rivals, a > 0is the reservation (or choke) price and s∈[0,1] measures the degree of substitutability between any pair of varieties. At any time during the game, the individual cost function is C i (t) = cq 2 i (t) + βk 2 i (t),(2) the first term accounting for extraction and production costs, the second for exploration costs, k i (t)≥0being the exploration effort. Function (2) says that all of the firm’s activities take place at decreasing returns to scale. Accordingly, firm i’s profit function is π i (t) = [p i −cq i (t)] q i (t)−βk 2 i (t).(3) In the remainder, I will investigate the two alternative scenarios in which firms either hold private rights on nsingle pools, one for each firm, or jointly (but noncooperatively) extract the resource from a single common pool. 2.1 Private pools Suppose each firm is allowed to explore its own drilling ground in order to expand it. The volume of resource existing at any tis x i (t),and its dynamics is described by the following state equation: · x i (t) = vx i (t)k i (t)−q i (t)(4) where v > 0is a constant common to all firms. It is worth noting that establishes that exploration is effective insofar as the stock has not been altogether exhausted. 4 Firm i’s current value Hamiltonian is H i (x,q,k) = e −ρt (p i −cq i (t)) q i (t)−βk 2 i (t) + λ i (vx i (t)k i (t)−q i (t)) (5) which the firm has to maximise w.r.t. controls q i (t)and k i (t),given the set of initial conditions x i0 =x i (0) >0.In (5), λ i (t) = µ i (t)e ρt is the ‘capitalised’ costate variable, while ρ > 0is the constant discount rate, common to all firms. The solution concept is the open-loop Nash equilibrium. 4 The necessary conditions are ∂H i (·) ∂q i =a−2 (1 + c)q i −sQ −i (t)−λ i = 0 (6) ∂H i (·) ∂k i =λ i vx i −2βk i = 0 (7) · λ i = (ρ−vk i )λ i (8) and the transversality condition is lim t→∞ e −ρt λ i x i = 0 for each firm. At this point, observe that (8) admits the solution λ i = 0 at all times; this, if substituted back into (7), implies k i = 0 at all times as well. That is, there exist a solution driving the industry to exploit the natural resources without caring about the ultimate consequence, i.e., exhaustion. This depicts a perspective in which firms, as in most of the environmental problems we are accustomed with, do not internalise the consequences of their activities if regulation policies are absent or assumed away. This, more often than not, is the only possibility when an appropriate policy is not introduced. However, in the present model there exist an alternative and much more productive route that firms can take spontaneously, and it is the following. From (7), we obtain the expression for the optimal value of the costate variable: λ i =2βk i vx i (9) 4 The technical reason for this choice is that the model does not take a linear-quadratic form, or any other form for which an obvious candidate for the value function to be used under feedback information is available (see Dockner et al. 2000, ch. 7). There are, however, sound economic arguments that can be invoked to corroborate the adoption of open-loop rules in dynamic games of resource extraction (see Reinganum and Stokey, 1985; and Mohr, 1988). 5 and the control dynamics: · k i = v · λ i x i + · λ i x i  2β.(10) Imposing symmetry across states and controls, and using (9), (6) rewrites as follows: a−[2 (1 + c) + s(n−1)] q−2βk vx = 0,(11) which delivers the optimal Cournot-Nash output 5 q N =avx −2βk v[2 (1 + c) + s(n−1)] x(12) at every instant, including the steady state, with no need of deriving the kinematic equation of the individual quantity (i.e., in a quasi-static way). Then, (4) and (10) become · x=v 2 k[2 (1 + c) + s(n−1)] x 2 −avx + 2βk v[2 (1 + c) + s(n−1)] x;(13) · k=k[2βk −vx (a−r(2 (1 + c) + s(n−1)) x)] v[2 (1 + c) + s(n−1)] x 2 .(14) Imposing stationarity on the system (13-14), we obtain the coordinates of the steady state points in the space (x, k) : x ∗ P P =av ±a 2 v 2 −8βρ 2 [2 (1 + c) + s(n−1)] 2v[2 (1 + c) + s(n−1)] ρ;k ∗ PP =ρ v.(15) Note that a 2 v 2 >8βρ 2 [2 (1 + c) + s(n−1)] is necessary and sufficient for x ∗ ± ∈R + .In the remainder, I will assume this condition is satisfied. Before delving into any further analytical details of the game, we may pause to stress that (15) illustrates a striking but quite intuitive difference between the firms’ incentives when pollution and natural resources are, alternatively at stake, since profit-seeking agents will obviously tend to internalise the effects of their productive activities if these may ultimately jeopardise 5 Henceforth, superscript Nwill be used to indicate the Cournot-Nash output level, while starred values will refer to the steady state equilibrium. 6 their ability to extract surplus from consumers (which is more likely to be the case of natural resources than polluting emissions, all else equal). As a consequence, firms do invest positive amounts of resources in exploration even if - as here - they are not spurred to do so by any public policy. Back to the model, the solutions in (15) can be studied by linearising the system around the steady states, and examine the following 2×2Jacobian matrix: J=      ∂ · x ∂x ∂f · x ∂k ∂ · k ∂x ∂ · k ∂k       (16) whereby the stability properties of the state-control dynamics (13-14) depend on the sign and size of the trace T(J)and determinant ∆ (J)of the above Jacobian matrix: T(J) = 2βk −v[a−(2 (1 + c) + s(n−1)) (vk +ρ)x]x v[2 (1 + c) + s(n−1)] x 2 (17) ∆ (J) = k[2β(4vk −ρ)−v 2 (2a−ρ(2 (1 + c) + s(n−1)) x)x] v[2 (1 + c) + s(n−1)] x 2 (18) In correspondence of the ‘smaller’ solution, x ∗ P P − , k ∗ ,we have T(J)| x ∗ − = ρ > 0and ∆ (J)| x ∗ P P − ∝a 2 v 2 −8βρ 2 [2 (1 + c) + s(n−1)] +ava 2 v 2 −8βρ 2 [2 (1 + c) + s(n−1)] >0.(19) Moreover, it can be shown that ∆ (J)| x ∗ P P − >T(J)| x ∗ P P −  2 /4.Consequently, x ∗ P P − , k ∗ PP is an unstable focus. On the contrary, in correspondence of the ‘larger’ solution, x ∗ P P + , k ∗ P P , while we have again T(J)| x ∗ P P + =ρ > 0,we see that ∆ (J)| x ∗ P P + ∝a 2 v 2 −8βρ 2 [2 (1 + c) + s(n−1)] −ava 2 v 2 −8βρ 2 [2 (1 + c) + s(n−1)] <0,(20) which qualifies x ∗ P P + , k ∗ PP as a saddle point. 7 Proposition 4 In Model II, the aggregate investment in exploration monotonically increases in the size of firms’ population also when a common pool is exploited. Hence, it seems that the aggregate behaviour may reflect the Arrovian hypothesis if appropriate conditions hold. Comparing individual exploration efforts reveals that k ∗ CP −k ∗ P P ∝ −2 (n−1) 2a(n−1) + nβ −nβ [4a(n−1) + βn]<0 (50) so that exploiting a common pool leads firms (and the industry as a whole) to underinvest in research activity as compared to the case in which each firm enjoys private access to a separate resource pool. 4 Concluding remarks I have investigated two simple differential games based on alternative assumptions on demand and technology, in which Cournot firms, the lack of a resource policy notwithstanding, spontaneously internalise the consequences of profit-seeking behaviour on the residual stock of natural resources, and consequently activate costly exploration projects. In both settings, the foregoing analysis has shown that private access to a single pool creates higher incentives to invest in exploration than the joint exploitation of a common pool does. 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