MPRA Mnich Personal RePEc Archive Cyclical and constant strategies in renewable resorces extraction George Halkos University of Thessaly, Department of Economics November 0 Online at http://mpra.b.ni-menchen.de/34654/ MPRA Paper No. 34654, posted. November 0 9:4 UTC
Cyclical and constant strategies in renewable resorces extraction By George E. Halkos and George J. Papageorgio University of Thessaly, Department of Economics, Korai 43, 38333, Volos, Greece Abstract This paper is concerned with the classic topic of intertemporal resorce economics: the optimal harvesting of renewable natral resorces over time by one and several resorce owners with conflicting interests. The traditional management model, dating back to Plorde (970), is extended towards a two state model in which harvesting eqipment is treated as a stock variable. As a conseqence of this extension, an eqilibrim dynamics with bifrcations and limit cycles occr. Next we discss conflicts as a game with two types of players involved: the traditional fishermen armed with the basic eqipment and the heavy eqipment sers. Both players have a common depletion fnction, thoght as harvesting, which is dependent both on personal effort and on intensity of eqipment s sage. Keywords: Renewable resorces; exploitation of natral resorces; differential games. JEL classifications: C6, C6, Q3.
. Introdction Intertemporal economic problems can be formlated either as optimal management models or as dynamic games. A basic difference between the two types of formlation is that, in the former case there is only one decision maker, the reglator, i.e. there is only one strategically acting agent, while in the latter there are more than one strategically interacting agents, choosing their actions that determine the crrent and ftre levels of tility. Consider, for example a single stock of an exhastible or reprodctive resorce that is simltaneosly exploited by several agents that do not cooperate. Each agent chooses an extraction strategy to maximize the disconted stream of ftre tility. Then, the actions taken determine not only their tility levels bt also the level of the stock. There are several implications of the above formlation. First, the actions taken by agents determine the size of a single capital stock that flly describes the crrent state of the economic system. Second, if there is no mechanism that forces players to coordinate their actions, they will act strategically and play a non cooperative game. Third, the eqilibrim otcome will critically depend on the strategy spaces available to the agents. There is a wide choice of possible actions (strategies) taken by the players. They may choose a simple time profile of actions and pre-commit themselves to these fixed actions over the entire planning horizon. Players then se open loop strategies. Alternatively players might choose feedback or closed loop or Markov strategies conditioning their actions on the crrent state of the system and reacting immediately every time the state variable changes, hence they are not reqired to precommit. Here we expose an example of several agents strategically exploiting the same renewable
3 resorce, like a fish stock, in order to expose the difference between open loop and closed loop strategies. If fisheries se open loop strategies they specify a time path of fishing effort in the beginning of the game and commit themselves to stick to these preannonced actions over the entire planning horizon. Alternatively, if they se feedback strategies they choose decision rles that determine crrent actions as a fnction of crrent stock of the resorce. Feedback decision rles captre the strategic interactions present in a dynamic game. If a rival fishery makes a catch today that necessarily reslts in a lower level of the fish stock, the opponents react with actions that take this change in the stock into accont. In that sense closed loop strategies captre all the featres of strategic interactions. In these lines, the main contribtion of this paper relies on the reslts obtained in the Nash eqilibrim of the game for which the players compete having a common harvesting (depletion) fnction. In eqilibrim terms we find the relation between the players discont factors in order to ensre eqilibrim with limit cycles. The strctre of the paper is the following. Section reviews the existing relative literatre. The next section concerns the conflicts as dynamic game with two players and with a common harvesting fnction. The last section concldes the paper.. Literatre review In environmental economics vast literatre, one given important meaning is connected with the exploitation of natral resorces. According to this approach a regeneration fnction is involved, which is necessary to model the interactions between the natre and the hman activities. In an important model, Strobele (988) considers the whole environment as renewable natral resorce and the damage done
4 to natre is described by a downward shift in the regeneration fnction de to the indstrial waste emission. In the same, bt more restrictive, way Hannesson (98) compares the optimality of the monopolistic and social planning extractions, finding that the monopolistic standing optimal stock of the resorce (say the natre) may either be larger or smaller than nder the social planning. Strobele and Wacker (994) extend the one-specie exploitation to mltiple species in a predator prey model. They derive a modified golden rle of harvesting, applying optimal control theory. Their conclsions abot the modified golden rle in the steady state, is related with the additional prodctivity effects. Farmer (000), reconsidering Mormoras type overlapping generations model with renewable natral resorces, shows that there exists a non trivial stationary state which exhibits, by definition, intergenerational natral capital eqality. Finally, natral resorces harvesting differs from prodction. Renewable resorces economic literatre, based on the fondations of Gordon (954), Scott (955) and Smith (969), sggests particlar properties of the open access natral resorces which reqires tools of analysis beyond those spplied by elementary economic theory. Sch an appropriate tool is the optimal control theory and the se of differential eqations in dynamic systems (either in a continos or a discrete framework), which are of common se in most models that explain the optimal management of natral resorces extraction. These systems depend on more than one parameter that measres different economic and biological characteristics of the exploited resorce. So the strctral stability is a key point to stdy in order to explore whether the qalitative dynamical properties of the system persist when its strctre is pertrbed. In this context, the stdy of the strctral stability is the first step to follow the analysis of the system.
5 On the other hand, it is reasonable to consider the stock of any renewable resorce as a capital stock and treat the exploitation of that resorce in mch the same way as one wold treat accmlation of a capital stock. This has been done to some extent by Clark (973) and Clark and Mnro (975), whose papers contain a discssion of this point of view. However, the analysis is mch simpler than it appears in the literatre especially since the interaction between markets and the natral biology dynamics has not been made clear. Frthermore renewable resorces are commonly analyzed in the context of models where the growth of the renewable resorce examined is affected by two factors: the size of the resorce itself and the harvesting rate. This specification does not take into accont that hman activities other than harvesting may have an impact on the growth of the natral resorce (Levhari and Withagen, 99). Some externalities may arise in maximm sstained yield programs of replenishable natral resorce exploitation followed by two fndamental problems. The first is that the existence of a social discont factor (or interest rate) may case the maximm sstained yield program to be non-optimal (Plorde, 970). The second problem relates to many externalities which may be present in harvesting resorces. The most significant of these externalities is the stock externality in prodction. That is, there is a potential misallocation of inpts in the prodction of natral resorce prodct de to the fact that one inpt, the natral resorce, contribtes to prodction bt may not receive payment, as nobody owns the resorce. An analysis of the biomass harvesting (like fisheries) mst take into accont the biological natre of fndamental capital, the renewable resorce, and mst recognize the common property featre of land or sea, so it mst allow that the fndamental capital is the sbject of exploitation. The problem of fishing indstry has
6 been tackled by economists giving attention to the common property characteristics associated with both the open access and the lack of proper property rights to the fishery indstry (Gordon, 954; Bjøndal, 99). A nmber of existing stdies on fishery economics have paid attention to the form of properties: fll rights or no rights at all (Smith, 969; Plorde, 97). Both cases lead to niqe Nash non-cooperative otcomes with the social planner s otcome in the case of fll rights and the open access in the case of no rights. The latter is the reslt of the tragedy of commons (for discssion see Clark and Mnro, 975). 3. A differential game with a common harvesting fnction Let s denote by xt the instantaneos renewable resorce which is in common access at time t. Withot any harvesting taking place the stock of resorces grows according to the fnction gx, obviosly dependent on the resorce itself, for all satisfying the conditions g0 0, g x 0 for all x 0, K, g x 0, g x 0 x K,. In the proposed game we assme that two types of players are involved. First it is the renewable resorce extractors (players) acting with the traditional mode in the sense of Clark (Clark, 990). The latter means that they are armed with the basic eqipment, sally harvests only personally, bt there is a crowd of this type of players. Next are the commercial heavy eqipment sers with a lot of vessels sally acting as factories. Carrying ot harvesting is costly for the second type of players, e.g. damages in the available eqipment, payroll for working men, also redcing its financial capital. Considering now the depletion of the renewable resorce stock (the harvesting fnction), one can thoght that however, does not only depend on the intensive sage
7 t of the heavy eqipped player, bt is also inflenced by the other players overall effort t which act traditionally. We set as instrment variables the intensity of eqipment and the personal harvesting effort respectively i.e. for the heavy eqipped player (player type ) the intensity of the harvesting eqipment s sage traditional fishermen (players of kind ) its personal effort i t, both assmed nonnegatives t 0, t 0. i We denote the overall harvesting fnction by, t, and for, also depending on overall effort t t and on intensity as well. Combining the growth gx with the harvesting fnction, i i the state dynamics can be written as x g x,, x x0 Along a trajectory the non-negativity constraint is imposed, that is 0 t 0 x t 0 0 () () A higher intensity of harvesting eqipment sage (for player ) and also the effort of the crowd of traditionally acting fishermen (player ) certainly leads to stronger depletion of the renewable resorce, so it is enogh reasonable to assme that the partial derivatives of the harvesting fnction to be positive with respect to the parameters, i.e. 0, 0. Moreover the law of diminishing retrns is applied only for the type player s effort ndertaken, that is 0 and for simplicity we assme 0. Additionally, we assme that the Inada conditions, which garantee that the optimal strategies are nonnegative, holds tre, i.e. 0 0 lim,, lim, 0 lim, 0, lim, (3)
8 The tility fnctions the two players want to maximize are defined as follows: Player, the representative traditional fishermen, derive instantaneos tility, on one hand from its own harvesting prodct, bt their personal effort t gives rise to increasing and convex costs a, and on the other hand from the high stock of renewable resorce also denoted by the increasing fnction fnctional x. After all the present vale of payer s tility is described by the following Player, the heavy eqipped, enjoys tility t, (4) J e x a dt 0 xfrom the renewable resorce stock xt, bt also from the eqipment s intensity of se, which is described by the fnction. Utilities x and are assmed monotonically increasing fnctions with decreasing marginal retrns, that is x x 0, 0 and 0, 0. We also assme that the individally acting players overall effort has no impact on player s tility. So, player s tility fnction is defined, in additively separable form, as: t (5) J e x dt 0 3.. Periodic Soltions In this sbsection we explore whether periodic soltions are possible, starting with steady state and stability analysis of necessary conditions. As it is clear the problem can be treated as a differential game with two controls and one state. Corresponding Hamiltonians, optimality conditions and adjoint variables for the problem nder consideration are respectively:
9,, H x a g x H x g x, H a, 0 (6) H, 0 (7) H gx x x (8) H gx x x (9) where sbscripts denote player and player respectively for Hamiltonias H i and adjoints i i,. Steady state soltions for the state, adjoints and controls are soltions of the system of eqations:, gx x g x, 0, 0, a, 0., gx x 0 The Jacobian matrix of the system of optimality conditions is the following x x x,, x g x J g x x gx 0 x g x x 0 gx x This also gives: tr J g x and
0 det gx x gx, J g x g x g x g x x g x, According to Wirl (997) the existence of a pair of prely imaginary eigenvales reqires that the following conditions are satisfied: J J w J w J tr 0, det 0, 0, det tr where coefficient w is the reslt of the sm of the following determinants w, gx g x 0 gx g xx gx 0, gx g x x g x,, gx g xx g xx From now on the crcial condition for cyclical strategies (precisely for Hopf bifrcations to occr) is that w 0, w det tr J J This after simple algebraic calclations redces to x g,, g xx gx x (0)
3.. Specifications for the game We specify the fnctions of the game as follows: - A diffsion process for the renewable resorce growth fnction, that is g x rx x, - A Cobb Doglas type fnction for the harvesting, and - The tility fnction stemming from eqipment s intensive se of player in the form A. Note that the tility fnction with A 0 and 0, exhibits constant relative risk aversion in the sense of Arrow Pratt measre of risk aversion. All the other fnctions are left in a linear form, i.e. both tilities stemming from the existing renewable resorce stock are for player x x and for player x x, while player s effort cost in the linear fashion a a, as well. Note that all the involved coefficients, i.e. the intrinsic growth rate r and the slopes, and a are positive real nmbers, bt 0, and A 0 and 0, With the above specifications the following reslt holds tre., as already mentioned. Proposition A necessary condition for cyclical strategies in the game between traditionally acting and heavy eqipped players, as described above, is the heavy eqipped players are more impatient than the simple traditionally acting. Proof In Appendix
The intition behind proposition is straightforward. We start with a rather low and increasing intensity of eqipment sage on behalf of the heavy eqipped players. The traditionally acting players operate at a low effort, as well, becase the increasing effort incrs costs, bt they are worrying abot the renewable resorce level, conseqently for their jobs, by the reason of player s presences. Now sppose that the heavy eqipped react as a farsighted, he wold increase the eqipment s intensity only moderately and the dynamical system wold approach a stable steady state. Bt, de to their impatience they behave myopically and react by strongly increasing the intensity of their machines. At this time the crowd of the traditionally acting players, has only two choices: to loose their jobs or to increase their overall effort. Sppose that they stay in the harvesting increasing their overall effort. Bt the latter means that the combination of high intensity on behalf of the heavy eqipped and the higher effort on behalf the crowd leads to a strong redction of the renewable resorce stock. Bt the low level of the resorce stock is nprofitable for the heavy eqipped to work at a high intensity, therefore they have to decrease intensity and the cycle is closed. A new cycle starts again, possibly in another place becase of the stock s redction, bt with the same reslts also described. In or opinion the crcial point of this intitive explanation is that player s strategic variable lags behind player s strategic variable and both are lagged behind the state variable, the renewable resorce s stock x.
3 3.3 The linear example Let s now calclate the Nash eqilibrim of the harvesting differential game. The concept of open-loop Nash eqilibrim is based on the fact that every player s strategy is the best reply to the opponent s exogenosly given strategy. Obviosly, eqilibrim holds if both strategies are simltaneosly best replies. Following Dockner et al (000), we formlate the crrent vale Hamiltonians for both players, as follows,, H x a g x H x g x, The first order conditions, for the maximization problem, are the following system of differential eqations for both players: First, the maximized Hamiltonians are H a, 0 () H, 0 () and second the costate variables are defined by the eqations H gx x x (3) H gx x x (4) The Hamiltonian of player, H, is concave in the control as far as and is garanteed by the assmptions on the signs of the derivatives, i.e. 0, 0 and from the decreasing marginal retrns on the player s tilities, i.e. x 0, 0. Moreover, optimality condition (54) implies that the adjoint
4 variable is positive only if player s marginal tility costs, since a,,. exceeds the marginal We also assme linearity of the model. A linear poplation growth fnction, despite the critiqe as a fairly nrealistic model, is a good approximation for the exponential growth of the hman poplation since 900 (Mrray, 00). To be more precise we specify the following fnctions of the game to be in linear form: i. the renewable resorce s growth fnction in the form gx is the growth rate, x, where ii. the tility fnction, x, which stems from the high stock of the renewable resorce, in the form x x iii. the fnction that measres player s effort cost in the form t a All the constants involved are positive nmbers, that is,, a 0. From the second player s side, the fnctions maximized are specified as linear, i.e. the tilities arisen from the resorce stock and high intensity realizations are written as x xt and t respectively. After the above simplified specifications the canonical system of eqations () - () can be rewritten as follows H, a 0 (5) H, 0 (6) H x (7) H x (8)
5 and the limiting transversality conditions has to hold t lim e x t t 0, lim e x t t 0 (9) t t t The analytical expressions of the adjoint variables, solving eqations (8)- (9), are respectively:, t t e C t t e C (0) () In order the transversality conditions to be satisfied it is convenient to choose the constant steady state vales, and therefore the adjoint variables collapses to the following constants, () To ensre certain signs for the adjoints () we impose another condition on the discont rates, which claim that discont rates are greater than the resorce s growth, i.e. we impose the condition, i, ths, the constant adjoint variables have both positive signs. i The above condition seems to be restrictive bt can be jstified as otherwise optimal soltions do not exist. Indeed, choosing, player s discont rate to be lower than the resorce s growth rate, their objective fnctional becomes nbonded in the case they choose to carry ot no harvesting. Similarly, choosing player s discont rate lower than the growth rate the associated adjoint variable becomes a positive qantity in the long rn. As a shadow price is implasible to be positive for
6 optimal soltions, the above reasoning is sfficient for the assmption i, i,. Once the concavity of the Hamiltonians, with respect to the strategies, for both players is satisfied the first order conditions garantee its maximization. Now, we choose the harvesting fnction s, specification, i.e. the specification of the fnction that redces the renewable resorce. This fnction is depending on both effort and intensity. We choose a similar to Cobb Doglas prodction fnction specification, which is characterized by constant elasticities in the following form:, 0 The rest of this section is devoted to the calclations of the explicit formlas at the Nash eqilibrim. 3.4. Optimal Nash Strategies Applying first order conditions for the chosen specification fnction gives s a a, (3), (4) The combination of (3) and (4), sing the Cobb Doglas type of specification, reveals an existing interrelationship between strategies, that is * * * * * * a * * a, (5) Expression (5) now predicts the interrelationship between the players Nash strategies, for which the reslt of comparison between them is dependent on the constant parameters and on the constant adjoint variables, as well.
7 Sbstitting (5) back into (4) we are able to find the analytical expressions of the strategies, after the following algebraic calclations. Expression (4) now becomes: * a a and from the latter the analytical expressions for the eqilibrim strategies are derived in a more comparable form now: * a * a (6) (7) Frther sbstittions in the eqation of the resorce s accmlation, x x, yield the following steady state vale of the stock x SS a (8) We smmarize the above discssion in a proposition. Proposition. Assming the harvesting fnction to exhibit constant elasticity and all the other fnctions to be linear, then the harvesting game yields constant optimal Nash strategies. The analytical expressions of the strategies are given by (6) and (7) for the traditional fishermen and the heavy eqipped respectively. The steady state vale of the resorces stock is given by expression (8).
8 Proposition seems to be with a little economic meaning cased by the linearity of the paradigm. Bt the constancy of the reslting strategies can be seen in connection with the concept of time consistency, a central property in economic theory. Time consistency is a minimal reqirement for a strategy s credibility, bt in general open-loop strategies haven t the time consistency property by defalt, since these strategies are time, and not state dependent fnctions. Nevertheless, a constant strategy may be a time consistent one, since the crcial characteristic for time consistency, i.e. the independency of any initial state x 0, is met for the above constant strategies. 5. Conclsions In environmental economics the exploitation of renewable resorces it is a well overlooked field since the original model, dated back to Schaffer (994). As it is well known the analysis concentrates on the two basic factors that affect the fishing indstry, namely the size of the resorce itself and the rate of hman harvesting. The above specification does not take into accont any other hman activities which affect biomass, for example coastlines polltion. Concerning long rn eqilibrim, as it is well known, the simplest case of the saddle point type stability reqires only one characteristic of the renewable resorce s growth fnction, i.e. the negative growth. Bt even the spposition of negative growth is sfficient for the saddle-point stability, the local monotonicity is not implied i.e. transient cycles may occr. On the other hand harvesting management is not restricted in the traditional way of the renewable resorce extraction in the sense of one man show. Commercial harvesting often reqires investment and disinvestment in eqipment, and the
9 ndertaken decision to expand or to redce eqipment obeys onto the state variable which is the existing renewable resorce stock. Therefore, concerning harvesting, as a stock variable, eqilibrim dynamics becomes more complex, and therefore mch richer, also inclding saddle point stability. In the discssion, in the main paper maid, the dynamics of sch eqilibrim dynamics reveal cyclical policies as optimal strategies, bt from the above discssion only some conclsions has been drawn. The emphasis given in the paper is restricted on the stability properties of the indced nonzero sm game between two types of players, which share a common depletion fnction thoght as a harvesting. Precisely, the game set p between a crowd of weakly armed and a strongly armed player with a common depletion fnction yields an economic reslt, for which the discont rate plays the crcial role for periodic soltions. That is, the condition for periodic soltions is that the strongly eqipped player to be more impatient than the weakly. Finally, for the spplement linear example of the same game we compte the optimal Nash strategies for both players, which are constant expressions.
0 Appendix Proof of proposition With the specifications, given in sbsection 4., one can compte, g x r,,,,, a g x r x, x, x H H Combining 0, a a 0,. and. the optimal strategies take the following forms a,.. * * and the optimal harvesting becomes * *, with the following partial derivatives a a a a * *, a * *,.3,.4.5.6.7
Both derivatives.6,.7 are negatives de to the assmptions on the parameters, 0, and on the signs of derivates, that is which ensres the positive sign of the adjoints,. 0, 0, x 0, x 0, Condition w J J det now becomes tr gx gx g x, which after sbstitting the vales from yields (at the steady states).6,.7 and making the rest of algebraic maniplations, finally, gx gx 0 gx Where we have set g x.8 stemming from the adjoint eqation g x x, which at the steady states redces into x g x. Condition 0 w after sbstittion the vales from.6,.7 becomes, g x w g x 0 gx The division.8 by yields.9, gx x The sm g 0 gx.0.9+.0 mst be positive, ths after simplifications and taking into accont that, g x, we have: g x g x g x the strict concavity of the logistic growth g 0. and the reslt follows from
References Berck P. (98). Optimal management of renewable resorces with growing demand and stock externalities. Jornal of Environmental Economics and Management, 8: 05 7. Bjørndal T. (99). Management of fisheries as a common property resorce. Paper presented at the XXI Conference of the International Association of Agricltral Economics, Tokyo, Agst -9. Clark C. (973). Profit maximization and the extinction of animal species, Jornal of Political Economy, 8: 950 96. Clark C. (990). Mathematical Bioeconomics, nd, Wiley Interscience. Clark C.W. and Mnro G.R. (975). Economics of fishing and modern capital theory: a simplified approach, Jornal of Environmental Economics and Management, : 9 06. Dockner E., Jorgensen S., Long N.V. and Sorger, G. (000). Differential Games in Economics and Management Science, Cambridge University Press. Farmer K. (000). Intergenerational Natral Capital Eqality in an Overlapping Generations Model with Logistic Regeneration, Jornal of Economics, 7: 9 5. Gordon H.S. (954). The economic theory of a common property resorce, Jornal of Political Economics, 6: 4 4. Hannesson R. (983). A Note on Socially Optimm verss Monopolistic Exploitation of a Renewable Resorce, Jornal of Economics, 43: 63 70. Levhari D. and Withagen C. (99). Optimal Management of the Growth Potential of Renewable Resorces, Jornal of Economics, 3: 97 309. Mrray J.D. (00). Mathematical Biology I: An Introdction, 3 rd Edition, Springer. Plorde C.G. (970). A simple model of replenishable natral resorce exploitation, American Economic Review, 6: 58 5. Plorde C.G. (97). Exploitation of common property replenishable resorces, Western Economic Jornal, 9: 56-66. Schafer M. (994). Exploitation of natral resorces and polltion. Some differential game models, Annals of Operations Research, 54: 37 6. Scott A. (955). Natral resorces: The economics of conservation, University of Toronto Press, Toronto, Canada. Smith V.L. (969). On models of commercial fishing, Jornal of Political Economy, 77: 8 98.
3 Strobele W. (988). The Optimal Intertemporal Decision in Indstrial Prodction and Harvesting a Renewable Natral Resorce, Jornal of Economics, 48: 375 388. Strobele W. and Wacker H. (995). The Economics of Harvesting Predator Prey Systems, Jornal of Economics, 6: 65 8. Wirl F. (997). Stability and limit cycles in one dimensional dynamic optimizations of competitive agents with a market externality, Jornal of Evoltionary Economics 7: 73 89.