The voluntary-threat approach to control nonpoint source pollution under uncertainty

•The voluntary-threat approach of Segerson and Wu (2006) is extended to the stochastic case.•The threat of a tax payment induces the polluters to engage in abatements at the optimal level.•Fine-tuning the cutoff value of the tax payment avoids the actual imposition of the tax.•The regulator may also randomize the tax imposition even if the cutoff level is exceeded.

This paper extends the voluntary-threat approach of Segerson and Wu (2006) to the case that the ambient level of nonpoint source pollution is stochastic. It is shown that when the random component is bounded from the above, fine-tuning the cutoff value of the tax payments avoids the actual imposition of the tax while the threat of such payments retains necessary incentive for the polluters to engage in abatements at the optimal level. If the random component is not bounded, the imposition of the tax cannot be completely avoided but the probability can be reduced by setting a higher cutoff value. It is also noted that the regulator has additional flexibility in randomizing the tax imposition but the randomization process has to be credible.