Regulating the tragedy of commons: Nonlinear feedback solutions of a differential game with a dual interpretation

Lambertini, Luca Working Paper Regulating the tragedy of commons: Nonlinear feedback solutions of a differential game with a dual interpretation Quaderni - Working Paper DSE, No. 1096 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Lambertini, Luca (2017) : Regulating the tragedy of commons: Nonlinear feedback solutions of a differential game with a dual interpretation, Quaderni - Working Paper DSE, No. 1096, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/5519 This Version is available at: https://hdl.handle.net/10419/177596 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/ ISSN 2282-6483 Regulating the tragedy of commons: nonlinear feedback solutions of a differential game with a dual interpretation Luca Lambertini Quaderni - Working Paper DSE N°1096 Regulating the tragedy of commons: nonlinear feedback solutions of a differential game with a dual interpretation 1 Luca Lambertini Dipartimento di Scienze Economiche Università di Bologna Strada Maggiore 45, 40125 Bologna, Italy [email protected] March 9, 2017 1 I would like to thank the seminar audience at the Department of Economics of the University of Salento for helpful comments and discussion. The usual disclaimer applies. Abstract A well established dynamic model describing the impact of oligopolistic interaction on a renewable resource is revisited here to illustrate its dual interpretation as a waste removal differential game. The regulatory implications are illustrated by assuming that the public agency may control market price and possibly also access to the commons. Two different formulations of the managerial or CSR objective are envisaged, based on a combination of profits and either output or the individual share of the waste stock. It is shown that if the representative firm’s objective includes the residual waste stock, there exists a unique regulated price driving to zero the steady state stock itself. Hence, the present analysis delivers some useful indications concerning an appropriate definition of the CSR objective firms should adopt. JEL Codes: C73, L13, Q20, Q53 Keywords: waste removal; resource extraction; feedback information; regulation; tragedy of commons 1 Introduction Free access to the commons is the driver of the original formulation of the tragedy in Gordon (1954) and Hardin (1968). This, in terms of oligopoly games, directly translates into the question as to whether there might exist an optimal industry structure, or, an optimal number of firms in the commons. The analysis of this problem can be traced back to Cornes et al. (1986), Mason et al. (1988) and Mason and Polasky (1997, 2002). What follows presents in a single model the impact of oligopolistic interaction on a renewable resource and a waste stock via a differential game approach. The idea that originated this paper stems from an elementary analogy between the exploitation of a renewable natural resource and waste removal, provided the dynamics according to which these two magnitudes grow over time can be assumed to be exogenously given and identical. The issue at stake, then, boils down to the following: if the state is a natural resource or species, in line of principle it would be desirable to have the largest possible stock of it left at the steady state, while the opposite holds if the state variable consists of waste. Hence, the policy implications of the ensuing analysis will be opposite in the two cases. In building up the model, I will pose that firms define their individual objective functions attaching a positive weight to their output levels or harvest rates or, alternatively, to the individual symmetric share of the stock. That is, a firm’s objective function is defined as a combination of profits and either the control or the state variable. One way or the other, this approach, in the light of the typical interpretation deriving from an established view in the theory of industrial organization, amounts to saying that firms have separated ownership from control via delegation contracts to managers à la Vickers (1985). However, also in this respect one can spot a dual nature of this additional feature, whereby if the common pool is a stock of waste then 1 maximising a combination of profits and output reveals the adoption of a CSR stance by the same firms. 1 For the sake of simplicity, in the remainder I will quite freely refer to the state variable as a renewable resource or a waste stock, and specify the relevant interpretation of the state when it comes to evaluating the consequences of firms’ behaviour, and therefore also the design of an appropriate regulation. In particular, if the state measures a stock of waste, the ensuing analysis shows that including the state in the maximand is definitely preferable to the alternative based on a combination of profits and individual output (or waste removal). This is because under this specification of the model the regulator avails of a unique regulated price which drives to zero the residual stock associated to any stable equilibria arising under feedback information. The structure of the paper is the following. The basic setup is laid out in section 2. The first version, where the CSR or managerial objective features the output level, is fully characterised in section 3, including the unregulated open-loop, linear and nonlinear feedback solutions as well as the regulated feedback game. Section 4 accounts for the linear and nonlinear feedback solutions of the alternative model, for the regulated case only. Plausible extensions and concluding remarks are in section 5. 2 The setup The setup is an extension of Lambertini and Mantovani (2014) and Benchekroun (2008), where a common property productive asset oligopoly is considered, and encompasses the duopoly model used in Benchekroun (2003) and Fuji1 This analogy between strategic delegation and corporate social responsibility has already beeen highlighted in the literature. See Laambertini and Tampieri (2015) and the references therein. 2 wara (2008). The model illustrates a differential oligopoly game of resource extraction unravelling over continuous time t∈[0,∞).The market is supplied by n≥1firms 2 producing a homogeneous good, whose inverse demand function is p=a−Qat any time t, with Q= n i=1 q i . Firms share the same technology, characterised by the cost function C i =cq 2 i in which parameter c∈(0, a)is constant over time. Firms operate without any fixed costs. During production, each firm exploits a renewable natural resource, whose accumulation is governed by the following dynamics: · S=F(S)−Q(1) with F(S) =          δS ∀S∈(0, S y ] δS y S max −S S max −S y ∀S∈(S y , S max ] (2) where Sis the resource stock, δ > 0is its implicit growth rate when the stock is at most equal to S y and δS y is the maximum sustainable yield. Taken together, (1-2) imply that (i) if the resource stock is sufficiently small the population grows at an exponential rate; and (ii) beyond S y , the asset grows at a decreasing rate. Moreover, S max is the carrying capacity of the habitat, beyond which the growth rate of the resource is negative, being limited by available amounts of food and space. In the remainder, we will confine our attention to the case in which F(S) = δS. 3 Firms play noncooperatively and choose their respective outputs simultaneously at every instant. At t= 0,each firm hires a manager whose contract 2 Under monopoly the delegation to managers would not be operated by stockholders, but CSR could be adopted, so I’m intentionally not ruling out the monopoly case. Another good reason not to do so pops up in section 4. 3 As in Benchekroun and Long (2002), Fujiwara (2008) and Tornell and Velasco (1992), among several others. 3 specifies the instantaneous objective which the manager has to maximise. Delegation contracts are observable. As in Vickers (1985), the delegation contract establishes that the instantaneous objective function of manager i is a linear combination of profits and output: 4 M i =π i +θq i (3) in which θdetermines the relevance of output in the firm’s objective. An alternative approach consists in supposing that the CSR managerial incentive is M i =π i −θ·S n(4) where θis a weight attached to the individual symmetric share of the stock. Intuitively, θ > 0seems appropriate if the state is a stock of waste. In both cases, θis treated as a constant and is symmetric across the population of firms. The i-th manager maximises the following discounted payoff flow Ω i = ∞ 0 M i e −ρt dt, (5) under the constraint posed by the state equation · S=δS −Q(6) Parameter ρ > 0is the discount rate, common to all managers and constant over time. Obviously, if θ= 0,firms behave as pure profit-seeking entrepreneurial units. The analysis will be carried out under the following assumption: 4 This contract is equivalent to that considered in Fershtman and Judd (1987) and Sklivas (1987), where the maximand is a weighted average of profits and revenues, M i = απ i + (1 −α)R i , R i =pq i . A proof of the equivalence is in Lambertini and Trombetta (2002). 4 Assumption 1 δ > ρ [n(n+ 2c) + 1] /[2 (1 + c)] . This guarantees the positivity of the residual resource stock at the steady state under any feedback rules. That is, in the remainder I will leave the possibility of resource exhaustion due to an excessively large number of firms out of the picture, in order to focus solely on the effects of delegation. In the remainder of the paper, I will refer to the game relying on (3) as model I, while that using (4) will be model II. 3 Model I Here, delegation (or the adoption of a CSR stance) has the same structure as in Vickers (1985), the instantaneous managerial objective being (3). A few words will suffice to capture the essence of the open-loop solution, which, for several reasons, is of limited interest. In the remainder of this section, I will pose σ≡a+θfor the sake of simplicity. If firms don’t internalise the consequences of their behaviour at any time and play the individual (static) Cournot-Nash output q CN =σ n+ 1 + 2c(7) at all times, then the residual amount of the natural resource in steady state is S CN =nσ/ [δ(n+ 1 + 2c)] = Q CN /δ. As the remainder of the analysis is about to show, it is worth noting that the static solution corresponds to the open-loop steady state one, which in this game is unstable (see below). Let the initial condition be S(0) = S 0 >0. The relevance of the size of S 0 on the final resource stock as well as on the stability of solutions will be discussed in the ensuing analysis. 5 by the definition of σ. This boils down to the following: Proposition 2 The separation between ownership and control via delegation contracts based on output expansion enlarges the set of stable nonlinear feedback solutions. In particular, since ∂S NLT /∂θ > 0,the above proposition is accompanied by a relevant corollary: Corollary 3 The adoption of managerial incentives based on output expansion increases the upper bound of the SNLS set. This result can be rephrased to say that these particular type of managerial incentives allows for a larger stock of the resource surviving in correspondence of a nonlinear feedback solution, and prompts for the analysis of the so-called voracity effect (Lane and Tornell, 1996; Tornell and Lane, 1999), which can be briefly summarised as follows. In line of principle, one would expect that the higher the resource growth rate is, the higher should be the volume of that resource in steady state. However, this may not hold true as firms respond to any increase in the growth rate by hastening resource extraction, whereby one observes that ∂S/∂δ < 0in steady state, at least for sufficiently high levels of δ. The arising of such voracity effect has been highlighted, with pure profit-seeking units, in Benchekroun (2008) and Lambertini and Mantovani (2014). As in Lambertini and Mantovani (2014, p. 121), also here it can be easily shown that under linear and nonlinear feedback information the voracity effect operates. Take the weighted average of S LF and S NLT : S=φS LF + (1 −φ)S NLT (27) with φ∈[0,1] .There emerges that ∂S/∂δ < 0for sufficiently high levels of the growth rate δ, for any φ∈[0,1] ,thereby including the extremes of 12 the relevant interval of resource stock volumes in steady state. However, this property, combined with Lemma 1, Proposition 2 and Corollary 3, entails Proposition 4 Managerial incentives allowing for output expansion soften the voracity effect over the entire interval of nonlinear feedback solutions SNS. It would be tempting to interpret this conclusion as implying a beneficial effect of managerialization on resource preservation (or, an undesirable effect upon waste removal, in which case voracity is most welcome for intuitive reasons). However, this would be hazardous as the same issue should indeed be reassessed in presence of alternative incentive schemes, based for instance on market shares (Jansen et al., 2007; Ritz, 2008) or comparative performance evaluation (Salas Fumas, 1992; Miller and Pazgal, 2001). Yet, the possibility that delegating control to agents interested in expanding production might ultimately mitigate the pressure on the resource is a striking and unexpected feature of the present model. This fact finds its explanation in the multiplicative effect of this form of delegation on equilibrium outputs and the resource stock, as the delegation parameter θappears in market size σand makes it larger as seen from the managers’ standpoint. Since σis a at the same time a measure of profitability or demand level, this type of delegation (i) increases the maximum mark-up from a−cto a−c+θor equivalently (ii) shifts the demand upwards by θ. Consequently, the managerial inclination to expanding output is routed in the direction of affecting the mark-up level and this mechanism operates as a partial remedy to voracity in the range where the latter takes place. Therefore, albeit with some caution, this design of delegation contracts - admittedly, far from being general - is of public interest because it couples the usual elements connected with consumer surplus and profits with additional motives (perhaps more far-reaching) dealing with the impact of the separation between ownership and control on resource (and 13 species) preservation. Moreover, there remains the open question as to how a public agency could regulate access to the commons, in presence of a single stable linear feedback equilibrium and infinitely many stable nonlinear feedback equilibria. A plausible solution is proposed in the next section. 3.3 The regulated case The model remains the same as for the resource dynamics (6) and firms’ technology. Instead, here the price pis exogenously given, being a policy instrument in the hands of a public authority in charge of regulating access to the common resource pool. Accordingly, firm i’s instantaneous maximand writes M i (t) = [p−cq i (t) + θ]q i (t).(28) The problem is formally defined as above, as firm i’s HJB equation is ρV i (S) = max q i M i +∂V i (S) ∂S ·(δS −Q)(29) Solving the game on the basis of the same procedure (or equivalently using the method of the undetermined parameters), one obtains the following pair of strategies: q OL p =σ p 2c;q LF p =2cδ (2δ−ρ)S−(δ−nρ)σ p 2cδ (2n−1) (30) where (i) σ p ≡p+θ;(ii) superscripts have the same meaning as above; and (iii) subscript pindicates that the price of the final good is being regulated. While q OL p >0over the entire parameter space, q LF p >0for all 7 S > (δ−nρ)σ p 2cδ (2δ−ρ)>0(31) 7 La demonstration that indeed (δ−nρ)σ 2cδ (2δ−ρ)>0 14 The interesting implication of the price regulation is that, irrespective of the information structure underpinning firms’ strategies, the residual steady state resource stock is exactly the same: S p =nq LF,OL p δ=nσ p 2cδ (32) which amounts to saying the following: Proposition 5 Regulating price eliminates the multiplicity of stable feedback equilibria, with the single linear feedback one surviving. Moreover, (32) has two relevant implications that should equally attract the attention of the authority: •S p monotonically increases in n: hence, the minimum residual stock obtains in correspondence of n= 1. Recalling the dual interpretation of the nature of S, this fact has completely opposite implications concerning the socially efficient access to the commons. •S p monotonically increases in σand therefore also in the extent of delegation, θ: this reveals that including the individual instantaneous harvest rates in the delegation contracts (or, adopting a CSR stance) might or mighty not mean good news from the regulator’s standpoint, again in view of the dual interpretation of the model as for the nature of the state variable. Be that as it may, the picture looks as in Figure 3, where again the arrows indicate the dynamics of the state Sand illustrate that q OL p is unstable while q LF p is stable. Therefore, although they seem to yield the same steady state, derives from the solution of the model in which firms are pure-profit-seeking agents (i.e., θ= 0) and price is endogenously determined via the linear demand function instant by instant. 15 open-loop and feedback information structures are not equivalent at all. In particular, the outcome engendered by q OL p can either drop to S= 0 or exceed S y ,depending on the initial stock, 8 while the volume of the long-run equilibrium state variable generated by q LF p is surely S p =nσ p /(2cδ). Figura 3 The regulated case   S(0,0) q · S= 0 q OL p q LF p S p               To close the discussion carried out in this section, let’s focus our attention onto the case in which Sis a stock of waste. If so, then monopoly is the 8 A peculiar and somewhat paradoxical feature of the case of waste removal is that if the initial stock is sufficiently low, firms might involuntarily drive to zero the residual stock under myopic open-loop rules. Of course it is also true that if the inital stock is large then the adoption of open-loop strategies might cause the waste stock to shoot up to plus infinity. 16 socially efficient structure, with S NLT  n=1 =S LF  n=1 (33) for obvious reasons, and S LF  n=1 −S p | n=1 =ac −θ−p(1 + c) 2δc (1 + c)>0(34) for all p < min 0,ac −θ 1 + c(35) which entails that waste removal in monopoly should be subsidised if θ > ac. 4 Model II Here, the contract based on (4) says that the firm attaches a negative weight to the residual individual share of waste at the symmetric equilibrium. I will focus on the regulated case only, as here - unlike what we have seen in model I - the continuum of stable feedback equilibria arising with an unregulated price survives the regime change. Hence, in what follows it is assumed that pis an instrument in the regulator’s hands. Additionally, for reasons which will become apparent below, I will confine myself to the case in which the state variable is a stock of waste (or, equivalently, the control of all firms has been delegated to CSR managers). Solving this game under feedback information yields infinitely many subgame perfect strategies. This seemingly undesirable feature is driven by the fact that the HJB equation is solved by the following two linear feedback strategies: q a II =np (δ−ρ)−θ 2cn (δ−ρ);q b II =nnpρ + 4cδ 2 S+ 2θ−δ(p+ 2cρS)−θ 2cnδ (2n−1) (36) 17 which produce two different values of the residual waste stock in steady state: S a II =np (δ−ρ)−θ 2cδ (δ−ρ);S b II =np (δ−nρ)−θ(2n−1) 2cδ (δ−nρ)(37) The resulting graph replicates the picture appearing in Figure 1, with analogous properties. In particular, also here the first (open-loop) solution is unstable, while the second is stable. Of course, there are infinitely many nonlinear solutions, a subset of which is stable. This is portrayed in Figure 4, which, except for labels, is the same as in Figure 2. Figure 4 Non linear solutions in the alternative model   S(0,0) q A B T · S= 0 q ′ (S)→ ±∞ q a II q b II S b II S a II S T                    Hence, it is evident that regulation does not deliver uniqueness if the delegation contract (or the CSR stance) chosen by firms relies on the residual stock of waste. Yet, at a closer look, this scenario is not as discouraging as it might look at first glance. To grasp the intuition why it is not so, observe 18 that S a II −S b II =(n−1) (2δ−ρ)θ 2cδ (δ−ρ) (δ−nρ)>0(38) for all θ > 0. This simple result can be formulated as follows: Lemma 6 If firms adopt a CSR stance based on a negative weight attached to the individual share of residual waste stock, the stable feedback solution yields a lower residual stock than the unstable (open-loop) one. This fact has several relevant implications: (i) any stable nonlinear solution is more desirable than the open-loop one; (ii) unlike what happens in models dealing with natural resource exploitation, here the voracity effect (Tornell and Velasco, 1992; Lane and Tornell, 1996; Tornell and Lane, 1999) combined with feedback information is indeed desirable; and, more importantly, both S T and S b II are monotonically decreasing in θ. That is, Proposition 7 Intensifying the CSR component in the firm’s objective function brings about a decrease in the residual waste shock in any stable steady state reached through feedback strategies. It is worth noting that this is the opposite of what happens in the previous model, where (32) measures the residual stock. Last but not least, one may verify that for any θ > 0there exists a unique level of the regulated price at which S b II = 0: pS b II = 0=θ(2n−1) n(δ−nρ)>0(39) with ∂p S b II = 0 ∂n =[δ+ 2n(n−1) ρ]θ n 2 (δ−nρ) 2 >0.(40) The same applies for any equilibrium generated by nonlinear feedback strategies, whose residual stock is  S=αS b II + (1 −α)S T , α ∈(0,1) ,as both S b II and S T are linear in p. Moreover, () implies that, in monopoly, S a II =S b II and 19 therefore the infinitely many nonlinear equilibria vanish. These last findings can be summarised as follows: Corollary 8 To any stable feedback solution is associated a single price driving to zero the residual waste stock at equilibrium. This price takes its minimum value in monopoly, where the stable linear solution is the only one being relevant as the continuum of nonlinear equilibria disappears. This suggests that the regulator may indeed rely on a single firm, granting it the lowest price identified by pS b II = 0 n=1 . This simultaneously solves the problem associated with the multiplicity of equilibria and ensures full removal at the lowest cost for society. Of course we are not compelled to take this conclusion literally, in the sense that the correct implementation of the price pS b II = 0requires knowing the exact values of the set of parameters {n, δ, ρ, θ}.However, the sign of partial derivatives (39-40) represents a reliable qualitative indication for the regulator. 5 Concluding remarks In a nutshell, the foregoing analysis has shown that the acquired model describing the dynamic exploitation of a common pool renewable resource could be reinterpreted as a game of waste removal, changing a few labels. Of course, this involves a non trivial change of perspective, in particular when it comes to the need of regulating an oligopoly game generating a continuum of feedback equilibria. Firms are either managerial or CSR entities - depending on the interpretation being chosen - and their objective been defined in two different ways. The first formulation stipulates that the relevant objective contains profits and output (or, the instantaneous individual volume of waste removal). In this case, the adoption of feedback information generates a continuum of stable subgame perfect equilibria. The choice of regulating price 20 sweeps away the continuum of equilibria engendered by nonlinear strategies, leaving the regulator with a single stable linear feedback equilibrium whose performance depends on the price level and the number of firms being granted access to the commons. Hence, there appears that, combining appropriately price and entry regulation, the public authority can indeed outperform the most favourable unregulated feedback equilibrium in terms of the residual resource stock at the steady state. The second formulation assumes that managerial or CSR incentives are based on a combination of profits and the firm’s individual share of the residual stock of the state variable. If the latter measures the volume of waste, the model shows that, in correspondence of any stable feedback equilibrium, there exists a price at which the residual stock is indeed nil. Moreover, such a price decreases monotonically in the number of firms. Hence, the regulator may restrict access to a single firm and adopt the lowest of all such prices, thereby attaining the desired goal at the lowest possible tariff. Needless to say, the foregoing material does not exhaust the analysis of this topic. In addition to the obvious extensions accounting for the aforementioned alternative delegation contracts based upon market shares (Jansen et al., 2007; Ritz, 2008) or comparative performance evaluation (Salas-Fumas, 1992; Miller and Pazgal, 2001), a plausible and promising one is that in which either control variables or the stock implies polluting emissions. The first possibility is plausible if the state refers to a natural resource, and production based on harvest is polluting the environment; the second is intuitively related to a scenario in which the state variable is a stock of waste. This extension would enrich the currently scant literature modelling the simultaneous presence of resource extraction (or stock removal) and environmental damage or global warming (cf. Lambertini and Leitmann, 2013; and Lambertini, 2016b). 21