Abstract
Accounting for the stochastic nature of environmental outcomes when quantifying economic environmental trade-offs with mathematical programming models requires the use of probabilistic programming approaches like the upper partial moment (UPM) method. Application of the UPM model may result in overregulation and losses in farm profit because the probabilistic constraint is satisfied at a higher level than the specified compliance probability, resulting in conservative responses from polluters. The main objective of this article was to present the upper frequency method as an alternative to enforce a probabilistic constraint with a close bound to the actual compliance probability. The UFM uses binary variables in a linear programming framework to enforce the probability bound on an empirically distributed outcome variable. Results showed that the UPM model was very conservative in the estimation of the upper probability bound, which resulted in an overestimation of abatement costs and an underestimation of the average amount of pollution above the environmental goal. Inconsistencies also exist between the ranking of alternatives when comparing the UPM and UFM methods. The UFM is general enough to ensure that the technique can be applied to any problem where the researcher is concerned with the risk of exceeding a specified target level.
Highlights
Weersink et al (2002) argue that optimal resource allocation is important because of its effects on farm income and because of its environmental impact
The main conclusion from this research is that the upper partial moment (UPM) method of enforcing probabilistic constraints is very conservative as is evident from the comparison with the newly developed upper frequency method (UFM)
The UPM method underestimates the actual probability with which the environmental goal is achieved as indicated in the objective function value of the model and the degree by which the environmental goal is exceeded
Summary
Weersink et al (2002) argue that optimal resource allocation is important because of its effects on farm income and because of its environmental impact. Nonpoint source (NPS) pollution stemming from agricultural practices is seen as a major cause of the remaining waterquality problems in developed and developing countries (Shortle et al 1998; Rossouw and Gorgens 2005; Ranga Prabodanie et al 2010; Li et al 2014a, b). There is increased pressure on agriculture to use resources optimally in order to reduce the negative environmental effect caused by agricultural practices (Shortle et al 2001). Generating economic-environmental trade-off curves is a complicated endeavor and requires quantifying the interrelationships between sustainability indicators implied by the underlying biophysical processes and producers’ economic behavior (Ranga Prabodanie et al 2010).
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