Dynamic Modeling of the Harvesting Function: The Conflicting Case

George Emm Halkos, George J. Papageorgiou

Laboratory of Operations Research, Department of Economics, University of Thessaly, Volos, Greece.

DOI: 10.4236/me.2014.57073                

Abstract

This paper is concerned with the classic topic of intertemporal resource economics: the optimal harvesting of renewable natural resources over time by one and several resource extractors with conflicting interests. The traditional management model, dating back to Plourde [1], is overlooked both in the simple case for which the resource stock is treated as a state variable and in the improved case for which the harvesting equipment is treated as a stock variable. As a result in the extended case, the equilibrium richer than the saddle point, with bifurcations and limit cycles, is possible. While the results of the enriched management case are consistent with the concept of the pulse fishing, as this concept is introduced by Clark [2]-[4], in the conflicting case the conditions, under which the richer limit cycle equilibrium occurs, are not enough investigated. Therefore, we discuss conflicts as a game with two types of players involved: the traditional fishermen armed with the basic equipment and the heavy equipment users. Both players have a common depletion function, considered as harvesting, which is dependent together on personal effort and on intensity of equipment’s usage.

Keywords

Renewable Resources, Exploitation of Natural Resources, Optimal Control, Differential Games

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Plourde, C.G. (1970) A Simple Model of Replenishable Natural Resource Exploitation. American Economic Review, 62, 518-521.
[2]	Clark, C. (1973) Profit Maximization and the Extinction of Animal Species. Journal of Political Economy, 81, 950-961. http://dx.doi.org/10.1086/260090
[3]	Clark, C.W. and Munro, G.R. (1975) Economics of Fishing and Modern Capital Theory: A Simplified Approach. Journal of Environmental Economics and Management, 2, 92-106. http://dx.doi.org/10.1016/0095-0696(75)90002-9
[4]	Clark, C. (1990) Mathematical Bioeconomics. 2nd Edition, Wiley Interscience, Hoboken.
[5]	Strobele, W. (1988) The Optimal Intertemporal Decision in Industrial Production and Harvesting a Renewable Natural Resource. Journal of Economics, 48, 375-388. http://dx.doi.org/10.1007/BF01227543
[6]	Hannesson, R. (1983) A Note on Socially Optimum versus Monopolistic Exploitation of a Renewable Resource. Journal of Economics, 43, 63-70. http://dx.doi.org/10.1007/BF01283884
[7]	Strobele, W. and Wacker, H. (1995) The Economics of Harvesting Predator-Prey Systems. Journal of Economic, 61, 65-81. http://dx.doi.org/10.1007/BF01231484
[8]	Farmer, K. (2000) Intergenerational Natural Capital Equality in an Overlapping Generations Model with Logistic Regeneration. Journal of Economics, 72, 129-152. http://dx.doi.org/10.1007/BF01676980
[9]	Gordon, H.S. (1954) The Economic Theory of a Common Property Resource. Journal of Political Economics, 62, 124-142. http://dx.doi.org/10.1086/257497
[10]	Scott, A. (1955) Natural Resources: The Economics of Conservation. University of Toronto Press, Toronto.
[11]	Smith, V.L. (1969) On Models of Commercial Fishing. Journal Political Economy, 77, 181-198. http://dx.doi.org/10.1086/259507
[12]	Levhari, D. and Withagen, C. (1992) Optimal Management of the Growth Potential of Renewable Resources. Journal of Economics, 3, 297-309. http://dx.doi.org/10.1007/BF01237184
[13]	Bjrndal, T. (1987) Production Economics and Optimal Stock Size in a North Atlantic Fishery. Scandinavian Journal of Economics, 89, 145-164. http://dx.doi.org/10.2307/3440061
[14]	Plourde, C.G. (1971) Exploitation of Common Property Replenishable Resources. Western Economic Journal, 9, 256-266.
[15]	Liski, M., Kort, P. and Novak, A. (2001) Increasing Returns and Cycles in Fishing. Resource and Energy Economics, 23, 241-258. http://dx.doi.org/10.1016/S0928-7655(01)00038-0
[16]	Wirl, F. (1995) The Cyclical Exploitation of Renewable Resource Stock May Be Optimal. Journal of Environmental Economics and Management, 29, 252-261. http://dx.doi.org/10.1006/jeem.1995.1045
[17]	Lewis, T. and Schmalensee, R. (1979) Non-Convexity and Optimal Harvesting Strategies for Renewable Resources. Canadian Journal of Economics, 12, 677-691. http://dx.doi.org/10.2307/134873
[18]	Dawid, H. and Kopel, M. (1997) On the Economically Optimal Exploitation of a Renewable Resource: The Case of a Convex Environment and a Convex Return Function. Journal of Economic Theory, 76, 272-297. http://dx.doi.org/10.1006/jeth.1997.2305
[19]	Dockner, E., Jorgensen, S., Long, N.V. and Sorger, G. (2000) Differential Games in Economics and Management Science. Cambridge University Press, Cambridge. http://dx.doi.org/10.1017/CBO9780511805127
[20]	Dockner, E. and Feichtinger F. (1991) On the Optimality of Limit Cycles in Dynamic Economic Systems. Journal of Economics, 53, 31-50. http://dx.doi.org/10.1007/BF01227014
[21]	Dockner, E. (1985) Local Stability Analysis in Optimal Control Problems with Two State Variables. In: Feichtinger, G., Ed., Optimal Control Theory and Economic Analysis, 2, North Holland, Amsterdam, 89-113.
[22]	Kuznetsov, Y. (1997) Elements of Applied Bifurcation Theory. Springer, Berlin.
[23]	Grass, D., Caulkins, J., Feichtinger, G., Trangler, G. and Behrens, D. (2008) Optimal Control of Nonlinear Processes. Springer, Berlin. http://dx.doi.org/10.1007/978-3-540-77647-5
[24]	Wirl, F. (1997) Stability and Limit Cycles in One-Dimensional Dynamic Optimizations of Competitive Agents with a Market Externality. Journal of Evolutionary Economics, 7, 73-89. http://dx.doi.org/10.1007/s001910050035
[25]	Schafer, M. (1994) Exploitation of Natural Resources and Pollution. Some Differential Game Models. Annals of Operations Research, 54, 237-262. http://dx.doi.org/10.1007/BF02031736