Computing uncertainty in the optimum nitrogen rate using a generalized cost function

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A Python package, “EONR”, was developed for computing the economic optimum nitrogen rate (EONR) and its profile-likelihood confidence intervals (CIs) under economic conditions defined by the user. This work was motivated by the need to improve nitrogen fertilizer recommendations using the maximum return to nitrogen approach, specifically to make it easier for researchers and other practitioners to calculate uncertainty and consider externalities to the cost function while computing the EONR. The “EONR” package fits yield response data to a re-parameterized quadratic-plateau model, which is generally accepted as the most appropriate model for describing yield response to nitrogen in maize (the package also supports the quadratic model). Although grain price and fertilizer cost are typically the only economic factors producers consider for determining the EONR, this package allows the user to also consider variable costs and/or externalities. A general cost function may be desired if the user wishes to consider costs to the farm operation (e.g., equipment, technology, labor, etc.) or environmental costs/penalties that may result from excess fertilizer application (e.g., water treatment or health costs that result from pollution) in addition to the traditional fertilizer to grain price ratio. In addition to the development of the “EONR” Python package, the objectives of this work were to: (i) design an algorithm that utilizes a general cost function for computing the EONR and its profile-likelihood CIs for any crop and (ii) clearly document the methodology and algorithms used. The “EONR” Python package can be downloaded from the Python Package Index (, and installation instructions, tutorials, and supplementary background information can be found in the online documentation (

Original languageEnglish (US)
Article number105030
JournalComputers and Electronics in Agriculture
StatePublished - Dec 2019


  • Economic optimum nitrogen rate (EONR)
  • Profile-likelihood confidence intervals
  • Python
  • Quadratic-plateau model
  • Social cost of nitrogen

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