A Characterization of the Optimal Management of Heterogeneous Environmental Assets under Uncertainty

ABSTRACT

The application herein involves the optimal management of renewable and nonrenewable resources within the context of a stochastic model of optimal control. By characterizing the two dimensional Bellman solution, three rules with respect to resource management are established. Within the context of coastal development, this analysis may help to explain why renewable resources may become increasingly vulnerable to random external shocks as nonrenewable resources are depleted. Although existence of an optimal closed form solution to the multi-sector Bellman model remains an open mathematical question, this analysis offers a characterization which can be applied to other scenarios in economics or finance in which two assets following stochastic processes interact.

The application herein involves the optimal management of renewable and nonrenewable resources within the context of a stochastic model of optimal control. By characterizing the two dimensional Bellman solution, three rules with respect to resource management are established. Within the context of coastal development, this analysis may help to explain why renewable resources may become increasingly vulnerable to random external shocks as nonrenewable resources are depleted. Although existence of an optimal closed form solution to the multi-sector Bellman model remains an open mathematical question, this analysis offers a characterization which can be applied to other scenarios in economics or finance in which two assets following stochastic processes interact.

Cite this paper

F. Raymond, "A Characterization of the Optimal Management of Heterogeneous Environmental Assets under Uncertainty,"*Theoretical Economics Letters*, Vol. 2 No. 5, 2012, pp. 502-510. doi: 10.4236/tel.2012.25093.

F. Raymond, "A Characterization of the Optimal Management of Heterogeneous Environmental Assets under Uncertainty,"

References

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[2] P. Dasgupta and G. Heal, “The Optimal Depletion of Exhaustible Resources,” Review of Economic Studies, Vol. 41, 1974, pp. 3-28.

[3] C. W. Clark and G. R. Munro, “The Economics of Fishing and Modern Capital Theory: A Simplified Approach,” Journal of Environmental Economics and Management, Vol. 2, No. 2, 1975, pp. 92-106. doi:10.1016/0095-0696(75)90002-9

[4] C. W. Clark, F. H. Clarke and G. R. Munro, “The Optimal Exploitation of Renewable Resource Stocks: Problems of Irreversible Investment,” Econometrica, Vol. 47, No. 1, 1979, pp. 25-47. doi:10.2307/1912344

[5] R. C. Merton, “Optimum Consumption and Portfolio Rules in a Continuous-Time Model,” Journal of Economic Theory, Vol. 3, No. 4, 1971, pp. 373-413. doi:10.1016/0022-0531(71)90038-X

[6] R. C. Merton, “An Asymptotic Theory of Growth under Uncertainty,” Review of Economic Studies, Vol. 42, No. 3, 1975, pp. 375-393. doi:10.2307/2296851

[7] S. Fischer, “The Demand for Index Bonds,” Journal of Political Economy, Vol. 83, No. 3, 1975, pp. 509-534. doi:10.1086/260339

[8] R. S. Pindyck, “Adjustment Costs, Uncertainty, and the Behavior of the Firm,” American Economic Review, Vol. 72, No. 3, 1982, pp. 415-427.

[9] S. K. Swallow, “Depletion of the Environmental Basis for Renewable Resources: The Economics of Interdependent Renewable and Nonrenewable Resources,” Journal of Environmental Economics and Management, Vol. 19, No. 3, 1990, pp. 281-296. doi:10.1016/0095-0696(90)90074-9

[10] R. S. Pindyck, “Uncertainty and Exhaustible Resource Markets,” Journal of Political Economy, Vol. 88, No. 6, 1980, pp. 1203-1225. doi:10.1086/260935

[11] S. Chen and M. Insley, “Regime Switching in Stochastic Models of Commodity Prices: An Application to an Optimal Tree Harvesting Problem,” Journal of Economic Dynamics and Control, Vol. 36, No. 2, 2012, pp. 201-219. doi:10.1016/j.jedc.2011.08.010

[12] C. Skiadas, “Robust Control and Recursive Utility,” Finance and Stochastics, Vol. 7, 2003, pp. 475-489. doi:10.1007/s007800300100

[13] M. Insley and K. Rollins, “On Solving the Multirotational Timber Harvesting Problem with Stochastic Prices: A Linear Complementarity Formulation,” American Journal of Agricultural Economics, Vol. 87, No. 3, 2005, pp. 735-755. doi:10.1111/j.1467-8276.2005.00759.x

[14] E. S. Phelps, “The Golden Rule of Accumulation,” American Economic Review, Vol. 51, No. 4, 1961, pp. 638-643.

[15] R. M. Solow, “Intergenerational Equity and Exhaustible Resources,” Review of Economic Studies, Vol. 41, 1974, pp. 29-45.

[16] R. Bellman, “Dynamic Programming,” Princeton University Press, Princeton, 1957.

[17] F. R. Chang, “Optimal Growth and Recursive Utility: Phase Diagram Analysis,” Journal of Optimization Theory and Applications, Vol. 80, No. 3, 1994, pp. 53-67. doi:10.1007/BF02207773

[18] F. R. Chang, F. R. and A. G. Malliaris, “Asymptotic Growth under Uncertainty: Existence and Uniqueness,” Review of Economic Studies, Vol. 54, No. 1, 1987, pp. 169-174. doi:10.2307/2297452

[19] M. I. Kamien and N. L. Schwartz, “Dynamic Optimization,” 2nd Edition, North-Holland, New York, 1991.

[20] A. G. Malliaris and W. A. Brock, “Stochastic Methods in Economics and Finance,” North Holland, New York, 1982.

[21] J. E. Marsden, “Elementary Classical Analysis,” W.H. Freeman and Company, San Francisco, 1974.

[1] H. Hotelling, “The Economics of Exhaustible Resources,” Journal of Political Economy, Vol. 39, No. 2, 1931, pp. 137-175. doi:10.1086/254195

[2] P. Dasgupta and G. Heal, “The Optimal Depletion of Exhaustible Resources,” Review of Economic Studies, Vol. 41, 1974, pp. 3-28.

[3] C. W. Clark and G. R. Munro, “The Economics of Fishing and Modern Capital Theory: A Simplified Approach,” Journal of Environmental Economics and Management, Vol. 2, No. 2, 1975, pp. 92-106. doi:10.1016/0095-0696(75)90002-9

[4] C. W. Clark, F. H. Clarke and G. R. Munro, “The Optimal Exploitation of Renewable Resource Stocks: Problems of Irreversible Investment,” Econometrica, Vol. 47, No. 1, 1979, pp. 25-47. doi:10.2307/1912344

[5] R. C. Merton, “Optimum Consumption and Portfolio Rules in a Continuous-Time Model,” Journal of Economic Theory, Vol. 3, No. 4, 1971, pp. 373-413. doi:10.1016/0022-0531(71)90038-X

[6] R. C. Merton, “An Asymptotic Theory of Growth under Uncertainty,” Review of Economic Studies, Vol. 42, No. 3, 1975, pp. 375-393. doi:10.2307/2296851

[7] S. Fischer, “The Demand for Index Bonds,” Journal of Political Economy, Vol. 83, No. 3, 1975, pp. 509-534. doi:10.1086/260339

[8] R. S. Pindyck, “Adjustment Costs, Uncertainty, and the Behavior of the Firm,” American Economic Review, Vol. 72, No. 3, 1982, pp. 415-427.

[9] S. K. Swallow, “Depletion of the Environmental Basis for Renewable Resources: The Economics of Interdependent Renewable and Nonrenewable Resources,” Journal of Environmental Economics and Management, Vol. 19, No. 3, 1990, pp. 281-296. doi:10.1016/0095-0696(90)90074-9

[10] R. S. Pindyck, “Uncertainty and Exhaustible Resource Markets,” Journal of Political Economy, Vol. 88, No. 6, 1980, pp. 1203-1225. doi:10.1086/260935

[11] S. Chen and M. Insley, “Regime Switching in Stochastic Models of Commodity Prices: An Application to an Optimal Tree Harvesting Problem,” Journal of Economic Dynamics and Control, Vol. 36, No. 2, 2012, pp. 201-219. doi:10.1016/j.jedc.2011.08.010

[12] C. Skiadas, “Robust Control and Recursive Utility,” Finance and Stochastics, Vol. 7, 2003, pp. 475-489. doi:10.1007/s007800300100

[13] M. Insley and K. Rollins, “On Solving the Multirotational Timber Harvesting Problem with Stochastic Prices: A Linear Complementarity Formulation,” American Journal of Agricultural Economics, Vol. 87, No. 3, 2005, pp. 735-755. doi:10.1111/j.1467-8276.2005.00759.x

[14] E. S. Phelps, “The Golden Rule of Accumulation,” American Economic Review, Vol. 51, No. 4, 1961, pp. 638-643.

[15] R. M. Solow, “Intergenerational Equity and Exhaustible Resources,” Review of Economic Studies, Vol. 41, 1974, pp. 29-45.

[16] R. Bellman, “Dynamic Programming,” Princeton University Press, Princeton, 1957.

[17] F. R. Chang, “Optimal Growth and Recursive Utility: Phase Diagram Analysis,” Journal of Optimization Theory and Applications, Vol. 80, No. 3, 1994, pp. 53-67. doi:10.1007/BF02207773

[18] F. R. Chang, F. R. and A. G. Malliaris, “Asymptotic Growth under Uncertainty: Existence and Uniqueness,” Review of Economic Studies, Vol. 54, No. 1, 1987, pp. 169-174. doi:10.2307/2297452

[19] M. I. Kamien and N. L. Schwartz, “Dynamic Optimization,” 2nd Edition, North-Holland, New York, 1991.

[20] A. G. Malliaris and W. A. Brock, “Stochastic Methods in Economics and Finance,” North Holland, New York, 1982.

[21] J. E. Marsden, “Elementary Classical Analysis,” W.H. Freeman and Company, San Francisco, 1974.