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The generation of unintended residuals when producing intended outputs is the key factor behind our serious problems with pollution. The way this joint production is modelled is therefore of crucial importance for our understanding and empirical efforts to change economic activities in order to reduce harmful residuals. Estimation of efficiency and productivity when producing both intended and unintended outputs has emerged as an important research strand. The most popular models in the field are based on weak disposability between the two types of outputs and null jointness introduced by Shephard. The purpose of the paper is to show that these model types are seriously flawed. An alternative model based on the production theory of Frisch introduces technical jointness for the case when the unintended output is unavoidable. The materials balance based on physical laws tells us that when material inputs are used unintended outputs are unavoidable. The modelling of joint production must therefore reflect this. A key feature is that the two types of outputs should be separated using different production relations. This facilitates estimating two independent frontiers and calculating efficiency scores and Malmquist productivity changes for the two types using a non-parametric DEA model.

The generation of unintended residuals in the production of intended outputs is the key factor behind our serious problems with pollution. The way this joint production is modelled is therefore of crucial importance for our understanding and empirical efforts to change economic activities in order to reduce harmful residuals. The materials balance tells us that residuals stem from the use of material inputs. The modelling of joint production must therefore reflect this. A multi-equation model building on the factorially determined multi-output model of classical production theory can theoretically satisfy the materials balance. Potentially complex technical relationships are simplified to express each of the intended outputs and the unintended residuals as functions of the same set of inputs. End-of-pipe abatement activity is introduced for a production unit. Introducing direct environmental regulation of the amount of pollutants generated an optimal private solution based on profit maximisation is derived. Serious problems with the single-equation models that have dominated the literature studying efficiency of production of intended and unintended outputs the last decades are revealed. An important result is that a functional trade-off between desirable and undesirable outputs for given resources, as exhibited by single-equation models, is not compatible with the materials balance and efficiency requirements on production relations. Multi-equation models without this functional trade-off should therefore replace single equation models. Extending the chosen multi-equation model to allow for inefficiency, three efficiency measures are introduced: desirable output efficiency, residuals efficiency, and abatement efficiency. All measures can be estimated separately using the non-parametric DEA model.