We present methods and tools that can be used to study dynamic environmental resource management in a spatial setting, to explore spatially dependent regulation, and to understand pattern formation. In particular, we present the maximum principle and its use in the context of the emerging frontier of applications of optimal control of diffusive transport processes to environmental and resource economics. We show how optimal spatiotemporal control induces pattern formation and how deep uncertainty with a spatial structure can be handled with spatial robust control methods. Finally, we show how models with diffusive transport can be extended to allow for long-range effects and more general transport mechanisms.


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Literature Cited

  1. Alexeev VA, Jackson CH. 2013. Polar amplification: Is atmospheric heat transport important?. Clim. Dyn. 41:533–47 [Google Scholar]
  2. Alexeev VA, Langen PL, Bates JR. 2005. Polar amplification of surface warming on an aquaplanet in “ghost forcing” experiments without sea ice feedbacks. Clim. Dyn. 24:7–8655–66 [Google Scholar]
  3. Armaou A, Christofides PD. 2001. Robust control of parabolic PDE systems with time-dependent spatial domains. Automatica 37:161–69 [Google Scholar]
  4. Asano T. 2010. Precautionary principle and the optimal timing of environmental policy under ambiguity. Environ. Resour. Econ. 47:2173–96 [Google Scholar]
  5. Athanassoglou S, Xepapadeas A. 2012. Pollution control with uncertain stock dynamics: when, and how, to be precautious. J. Environ. Econ. Manag. 63:3304–20 [Google Scholar]
  6. Bardi M, Capuzzo-Dolcetta I. 2008. Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations Boston: Birkhauser [Google Scholar]
  7. Benveniste LM, Scheinkman JA. 1982. Duality theory for dynamic optimization models of economics: the continuous time case. J. Econ. Theory 27:11–19 [Google Scholar]
  8. Boucekkine R, Camacho C, Fabbri G. 2013a. On the optimal control of some parabolic partial differential equations arising in economics. Work. Pap. 34, Aix-Marseille Sch. Econ.
  9. Boucekkine R, Camacho C, Fabbri G. 2013b. Spatial dynamics and convergence: the spatial AK model. J. Econ. Theory 148:2719–36 [Google Scholar]
  10. Boucekkine R, Camacho C, Zou B. 2009. Bridging the gap between growth theory and the new economic geography: the spatial Ramsey model. Macroecon. Dynam. 13:120–45 [Google Scholar]
  11. Brito PB. 2011. Global endogenous growth and distributional dynamics. Work. Pap., ISEG/Tech. Univ. Lisbon/UECE
  12. Brock W, Engström G, Xepapadeas A. 2014. Spatial climate-economic models in the design of optimal climate policies across locations. Eur. Econ. Rev. 69:78–103 [Google Scholar]
  13. Brock W, Xepapadeas A. 2008. Diffusion-induced instability and pattern formation in infinite horizon recursive optimal control. J. Econ. Dyn. Control 32:92745–87 [Google Scholar]
  14. Brock W, Xepapadeas A. 2010. Pattern formation, spatial externalities and regulation in coupled economic–ecological systems. J. Environ. Econ. Manag. 59:2149–64 [Google Scholar]
  15. Brock W, Xepapadeas A, Yannacopoulos A. 2014a. Robust control and hot spots in spatiotemporal economic systems. Dyn. Games Appl. 4:257–89 [Google Scholar]
  16. Brock W, Xepapadeas A, Yannacopoulos A. 2014b. Spatial externalities and agglomeration in a competitive industry. J. Econ. Dyn. Control. 42:143–74 [Google Scholar]
  17. Brock WA, Engström G, Grass D, Xepapadeas A. 2013. Energy balance climate models and general equilibrium optimal mitigation policies. J. Econ. Dyn. Control 37:122371–96 [Google Scholar]
  18. Brock WA, Xepapadeas A. 2006. Diffusion-induced instability and pattern formation in infinite horizon recursive optimal control. Work. Pap., Univ. Wis., Madison; Athens Univ. Econ. Bus. http://dx.doi.org/10.2139/ssrn.895682 [Crossref]
  19. Brock WA, Xepapadeas A, Yannacopoulos A. 2014c. Optimal agglomerations in dynamic economics. J. Math. Econ. 53:1–15 [Google Scholar]
  20. Brock WA, Xepapadeas A, Yannacopoulos AN. 2014d. Robust control of a spatially distributed commercial fishery. In Dynamic Optimization in Environmental Economics [Dynamic Modeling and Econometrics in Economics and Finance, Vol. 15], ed. E Moser, W Semmier, G Tragler, VM Veliov, pp. 215–41. Berlin/Heidelberg, Ger.: Springer-Verlag. doi:10.1007/978-3-642-54086-8_10
  21. Camacho C, Zou B. 2004. The spatial Solow model. Econ. Bull. 18:1–11 [Google Scholar]
  22. Camacho C, Zou B, Briani M. 2008. On the dynamics of capital accumulation across space. Eur. J. Oper. Res. 186:2451–65 [Google Scholar]
  23. Derzko N, Sethi S, Thompson GL. 1980. Distributed parameter systems approach to the optimal cattle ranching problem. Optim. Control Appl. Methods 1:13–10 [Google Scholar]
  24. Derzko N, Sethi S, Thompson GL. 1984. Necessary and sufficient conditions for optimal control of quasilinear partial differential systems. J. Optim. Theory Appl. 43:189–101 [Google Scholar]
  25. Desmet K, Rossi-Hansberg E. 2010. On spatial dynamics. J. Reg. Sci. 50:143–63 [Google Scholar]
  26. Desmet K, Rossi-Hansberg E. 2012. On the spatial economic impact of global warming. NBER Work. Pap. 18546
  27. El-Farra N, Christofides P. 2001. Integrating robustness, optimality and constraints in control of nonlinear processes. Chem. Eng. Sci. 56:51841–68 [Google Scholar]
  28. Fanning AF, Weaver AJ. 1996. An atmospheric energy-moisture balance model: climatology, interpentadal climate change, and coupling to an ocean general circulation model. J. Geophys. Res. 101:D1015111–15 [Google Scholar]
  29. Gilboa I, Schmeidler D. 1989. Maxmin expected utility with non-unique prior. J. Math. Econ. 18:2141–53 [Google Scholar]
  30. Glowinski R, Lions J-L, He J. 2008. Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach Cambridge, UK: Cambridge Univ. Press [Google Scholar]
  31. Goetz RU, Zilberman D. 2000. The dynamics of spatial pollution: the case of phosphorus runoff from agricultural land. J. Econ. Dyn. Control 24:1143–63 [Google Scholar]
  32. Goetz RU, Zilberman D. 2007. The economics of land-use regulation in the presence of an externality: a dynamic approach. Optim. Control Appl. Methods 28:121–43 [Google Scholar]
  33. Hansen LP, Sargent TJ. 2001a. Acknowledging misspecification in macroeconomic theory. Rev. Econ. Dyn. 4:3519–35 [Google Scholar]
  34. Hansen LP, Sargent TJ. 2001b. Robust control and model uncertainty. Am. Econ. Rev. 91:260–66 [Google Scholar]
  35. Hansen LP, Sargent TJ. 2003. Robust control of forward-looking models. J. Monet. Econ. 50:3581–604 [Google Scholar]
  36. Hansen LP, Sargent TJ. 2008. Robustness Princeton, NJ: Princeton Univ. Press [Google Scholar]
  37. Hansen LP, Sargent TJ, Turmuhambetova G, Williams N. 2006. Robust control and model misspecification. J. Econ. Theory 128:145–90 [Google Scholar]
  38. Hassler J, Krusell P. 2012. Economics and climate change: integrated assessment in a multi-region world. J. Eur. Econ. Assoc. 10:5974–1000 [Google Scholar]
  39. HilleRisLambers R, Rietkerk M, van den Bosch F, Prins HH, de Kroon H. 2001. Vegetation pattern formation in semi-arid grazing systems. Ecology 82:150–61 [Google Scholar]
  40. Judd KL. 1998. Numerical Methods in Economics Cambridge, MA: MIT Press [Google Scholar]
  41. Kamien M, Schwartz N. 1991. Dynamic Optimization: The Calculus of Variations and Optimal Control in Economics and Management Amsterdam: Elsevier [Google Scholar]
  42. Komornik V, Loreti P. 2005. Fourier Series in Control Theory New York: Springer [Google Scholar]
  43. Krugman PR. 1996. The Self-Organizing Economy Cambridge, MA: Blackwell [Google Scholar]
  44. Kyriakopoulou E, Xepapadeas A. 2013. Environmental policy, first nature advantage and the emergence of economic clusters. Reg. Sci. Urban Econ. 43:1101–16 [Google Scholar]
  45. Leitemo K, Söderström U. 2008. Robust monetary policy in the new Keynesian framework. Macroecon. Dyn. 12:Suppl. 1126–35 [Google Scholar]
  46. Maenhout PJ. 2004. Robust portfolio rules and asset pricing. Rev. Financ. Stud. 17:4951–83 [Google Scholar]
  47. Maenhout PJ. 2006. Robust portfolio rules and detection-error probabilities for a mean-reverting risk premium. J. Econ. Theory 128:1136–63 [Google Scholar]
  48. Magill MJ. 1977a. A local analysis of N-sector capital accumulation under uncertainty. J. Econ. Theory 15:1211–19 [Google Scholar]
  49. Magill MJ. 1977b. Some new results on the local stability of the process of capital accumulation. J. Econ. Theory 15:1174–210 [Google Scholar]
  50. Murray JD. 2002. Mathematical Biology, Volume 2. New York: Springer
  51. Nordhaus WD. 2010. Economic aspects of global warming in a post-Copenhagen environment. Proc. Natl. Acad. Sci. USA 107:2611721–26 [Google Scholar]
  52. North GR, Cahalan RF, Coakley JA. 1981. Energy balance climate models. Rev. Geophys. 19:191–121 [Google Scholar]
  53. Oksendal B. 2005. Optimal control of stochastic partial differential equations. Stoch. Anal. Appl. 23:1165–79 [Google Scholar]
  54. Onatski A, Williams N. 2003. Modeling model uncertainty. J. Eur. Econ. Assoc. 1:51087–122 [Google Scholar]
  55. Papageorgiou YY, Smith TR. 1983. Agglomeration as local instability of spatially uniform steady-states. Econometrica 51:41109–19 [Google Scholar]
  56. Petracou E, Xepapadeas A, Yannacopoulos A. 2013. The bioeconomics of migration: a selective review towards a modelling perspective. Work. Pap. 1306, Athens Univ. Econ. Bus
  57. Roseta-Palma C, Xepapadeas A. 2004. Robust control in water management. J. Risk Uncertain. 29:121–34 [Google Scholar]
  58. Smith MD, Sanchirico JN, Wilen JE. 2009. The economics of spatial-dynamic processes: applications to renewable resources. J. Environ. Econ. Manag. 57:1104–21 [Google Scholar]
  59. Turing A. 1952. The chemical basis of morphogenesis. Philos. Trans. R. Soc. 237:37–72 [Google Scholar]
  60. Vardas G, Xepapadeas A. 2010. Model uncertainty, ambiguity and the precautionary principle: implications for biodiversity management. Environ. Resour. Econ. 45:3379–404 [Google Scholar]
  61. Wald A. 1950. Statistical Decision Functions New York: Wiley [Google Scholar]
  62. Weaver AJ, Eby M, Wiebe EC, Bitz CM, Duffy PB et al. 2001. The UVic earth system climate model: model description, climatology, and applications to past, present and future climates. Atmos. Ocean 39:4361–428 [Google Scholar]
  63. Wilen JE. 2007. Economics of spatial-dynamic processes. Am. J. Agric. Econ. 89:51134–44 [Google Scholar]
  64. Wu W, North GR. 2007. Thermal decay modes of a 2-D energy balance climate model. Tellus A 59:5618–26 [Google Scholar]
  65. Xabadia A, Goetz R, Zilberman D. 2004a. Optimal dynamic pricing of water in the presence of waterlogging and spatial heterogeneity of land. Water Resour. Res. 40:7W07S02 [Google Scholar]
  66. Xabadia M, Goetz RU, Zilberman D. 2004b. Spatially and intertemporally efficient management of waterlogging. Tech. Rep., Dep. Econ., Univ. Girona
  67. Xepapadeas A, Yannacopoulos A. 2013. Climate change policy under spatially structured ambiguity: hot spots and the precautionary principle. Work. Pap. 1332, Athens Univ. Econ. Bus.
  68. Yannacopoulos AN. 2008. Rational expectations models: an approach using forward–backward stochastic differential equations. J. Math. Econ. 44:3251–76 [Google Scholar]
  69. Zuazua E. 2007. Controllability and observability of partial differential equations: some results and open problems. Handb. Differ. Equ. Evol. Equ. 3:527–621 [Google Scholar]

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