Abstract
Traditional means of studying environmental economics and management problems consist of optimal control and dynamic game models that are solved for optimal or equilibrium strategies. Notwithstanding the possibility of multiple equilibria, the models’ users—managers or planners—will usually be provided with a single optimal or equilibrium strategy no matter how reliable, or unreliable, the underlying models and their parameters are. In this paper we follow an alternative approach to policy making that is based on viability theory. It establishes “satisficing” (in the sense of Simon), or viable, policies that keep the dynamic system in a constraint set and are, generically, multiple and amenable to each manager’s own prioritisation. Moreover, they can depend on fewer parameters than the optimal or equilibrium strategies and hence be more robust. For the determination of these (viable) policies, computation of “viability kernels” is crucial. We introduce a MATLAB application, under the name of VIKAASA, which allows us to compute approximations to viability kernels. We discuss two algorithms implemented in VIKAASA. One approximates the viability kernel by the locus of state space positions for which solutions to an auxiliary cost-minimising optimal control problem can be found. The lack of any solution implies the infinite value function and indicates an evolution which leaves the constraint set in finite time, therefore defining the point from which the evolution originates as belonging to the kernel’s complement. The other algorithm accepts a point as viable if the system’s dynamics can be stabilised from this point. We comment on the pros and cons of each algorithm. We apply viability theory and the VIKAASA software to a problem of by-catch fisheries exploited by one or two fleets and provide rules concerning the proportion of fish biomass and the fishing effort that a sustainable fishery’s exploitation should follow.
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Notes
Viability theory applications in other economic areas include: finance—Pujal and Saint-Pierre (2006); managerial economics—Krawczyk et al. (2012); macroeconomics —Krawczyk and Kim (2009); Bonneuil and Saint-Pierre (2008); Bonneuil and Boucekkine (2008); Krawczyk and Kim (2004); Krawczyk and Sethi (2007); Clément-Pitiot and Saint-Pierre (2006); Clément-Pitiot and Doyen (1999); microeconomics—Krawczyk and Serea (2011). However, several of the above publications are working papers of limited circulation.
We note that constraining \(e(t)\) from below may be redundant given the constraint for profit \(\pi (t)\).
\(B_0\) is widely used in the literature to denote the virgin or unexploited biomass, which is equivalent to the carrying capacity in the logistic growth model.
For simplicity, the following specification is made, \(\delta ^-=-\delta ^+\), which means that the maximum increase in effort must be the same as the absolute maximum decrease.
In some situations, \(p_y\) could be negative to represent a penalty incurred by the fisherman for catching the by-catch species. This could though be a politically problematic solution because of underreporting or non-reporting and dumping possibilities of the stock \(y\).
Similar problems are classified as a qualitative games, see Cardaliaguet et al. (1999).
A calibration of the second fleet more in line with the first fleet’s produced a kernel with slices for each fleet that looked very similar, making its study in this paper less interesting.
Viability is normally defined in terms of an infinite time horizon, but it is also possible to define \(\Theta \equiv [0, T], T \in \mathbb {R}_+\), and talk about finite-time viability.
For existence and characterisation of feedback controls assuring viability see Veliov (1993).
If \(D\) were a disc, then a contingent cone at any point of the circumference would be a half-space. Also, when \(\bar{x}\) is an interior point of \(D\), then the contingent cone for this point is the whole space.
For instance, inflation-targetting central banks will often avoid changing interest rates for as long as they can.
There exist more algorithms for kernel computations, see papers referenced in footnote 1.
The discount rate \(\varrho >0\) is here a computational artefact needed for the policy improvement method implemented in InfSOCSol and does not need to be economically interpretable. Also, because of the curse of dimensionality we do not think \(n\) could be greater than 7 because exponentially more memory is required for each additional dimension.
It is determined that no optimal solution can be found when one of two things happens: either (a) the numerical optimisation routine employed by InfSOCSol throws an error whilst attempting to find a solution; or (b) the solution produced by InfSOCSol does not, when tested, succeed in maintaining the system within the constraints.
Symbol \(\min ^G\) refers to the numerical method of function minimisation employed. In MATLAB, the principal algorithm used is fmincon, available from the Optimization Toolbox. This algorithm is able to numerically solve non-linear constrained optimisation problems.
The “percentage viable” figures are computed relative to the total number of grid points considered in each discretisation, including points that are outside of the non-linear constraints (i.e. left of line \(A\)), and are given to four significant figures.
Its performance though depends on the tolerances used by InfSOCSol.
Note however, that an increase in the discretisation of a single state-space dimension will increase the computation time by an exponent of the overall number of dimensions due to the curse of dimensionality. Hence, increases in discretisation should be made judiciously.
Memory and matrix addressing limits are dependent on a number of factors other than CPU architecture. Refer to the MATLAB and/or Octave documentation for more information.
Under the logistic growth fisheries model, the level of biomass that supports the maximum sustainable yield is equal to half the carrying capacity (Schaefer 1954).
VIKAASA stands for Viability Kernel Approximation, Analysis and Simulation Application = VIKAASA, which happens to be a Sanskrit word
that means “progress” or “development”.
VIKAASA is also compatible with GNU Octave, but without the GUI. See the manual for more information.
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Acknowledgments
We are grateful to two anonymous referees for their insightful reports and improvement requests that have assisted us in clarifying and, hopefully, sharpening this paper’s message. We also thank Vlado Petkov for his comments on some parts of this paper presented during the 2013 New Zealand Macroeconomic Dynamics Workshop in Wellington. All remaining errors are ours.
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A. Pharo was funded by 2012–2013 VUW DVC Research Grant “Sustainability explained by economic theory and mathematical methods”, Award 3230, Project 200757.
Appendices
Appendix A: Model calibration
See Table 2.
Appendix B: VIKAASA
VIKAASA a is specialised MATLAB application that can compute viability kernel approximations for rectangular constraint and control sets.Footnote 23
VIKAASA can be used either as a set of MATLAB functions,Footnote 24 or via a graphical user interface (GUI), as in Fig. 14. Using the GUI one can specify the viability problem for which to kernel is sought, run the kernel approximation algorithms and view the results. VIKAASA also supports saving and viability kernel data into files.
VIKAASA main window
A detailed (although somewhat out of date) manual for VIKAASA can be found in Krawczyk and Pharo (2011). The latest version of VIKAASA is available for download at http://code.google.com/p/vikaasa/.
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Krawczyk, J.B., Pharo, A., Serea, O.S. et al. Computation of viability kernels: a case study of by-catch fisheries. Comput Manag Sci 10, 365–396 (2013). https://doi.org/10.1007/s10287-013-0189-z
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DOI: https://doi.org/10.1007/s10287-013-0189-z