Similar situations arise in the analysis of air quality. In both cases, water and air, there
are questions related to irreversible degradation of the environment. For example,
environmental damages due to economic activities may be irreversible, with the level at
which the degradation becomes irreversible unknown. Particular attention must be paid
to the situation where agents do not place a high priority on degradation of the
environment and/or regeneration of the environment occurs slowly. Optimal policy
decisions vary depending on whether irreversibility is considered and the behavior of
humans and the environment is uniform. It would be very useful to develop mathematical
theories for optimal decision-making under uncertainty which reflect constraints
regarding sustainability, degradation, regeneration and their interactions, and apply
these in the context of modeling water and air quality. Any criterion of optimality must
include the condition of sustainability.
The reserves of coal, oil, natural gas and uranium are limited. In addition,
products resulting from their use – e.g., carbon dioxide and radioactive waste -- cannot
be fully absorbed by the environment. Consequently these reserves are not sustainable
sources of energy, nor is ethanol from corn, which requires fossil-energy input for
plowing the fields and distilling the mash, as well as large quantities of water. In the long
run, the only sustainable energy is renewable energy, such as solar, wind and
hydropower. As demand for renewable energy increases, it becomes important to devise
optimal strategies to achieve given demands. Iniyan (1998) explores this issue from a
mathematical point of view. Some mathematical models show that sustainability (in
energy or other resources) can be achieved if compensation is possible (i.e., stocks for
renewable resources augmented as production depletes stocks of nonrenewable natural
resources). Developing mathematical models and analysis for dynamical network
models has promise to advance this area of sustainability science.
Re-use or recycling of natural resources is a key component of sustainability. So
could be strategies such as “cap and trade” that encourage trade-offs to slow depletion
of natural resources or production of unwanted byproducts of human processes. Mellor
et al. (2002) create a model of re-use of natural resources using a methodology that
considers multiple-use phases by describing material recovery, re-use, and recycling.
The model serves to generate a set of pareto-optimal choices needed to support multi-
attribute decisions in which technical, economic and environmental performances must
all be considered. A recent book by DeLara and Doyen (2008) offers a mathematically-
based course on trade-offs for sustainable management of natural resources. It
introduces mathematical models of viability; concepts such as decisions under
uncertainty; tools such as the Pontryagin maximum principle; maximum approaches;
robust control; and stochastic optimization. In addition, a new mathematical framework
for competitive equilibrium, in which emissions trading schemes can be analyzed, was
very recently introduced (Carmona et al. 2010).
The role of human institutions such as property rights and pricing systems for
natural resources is pivotal in achieving growth and improved distribution of income and
wealth, in understanding environmental degradation, and in seeking improved policy
leading toward sustainability. The role of such institutions in promoting sustainable
development is addressed using mathematical approaches by Veeman et al. (2003).
Special management problems exist for ‘critical’ components of natural capital to ensure
that our heirs receive an undiminished patrimony. It would be very useful to develop
concrete models that address specific human institutions. Particularly useful criteria and
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