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Energy, Environment and Transitional Green Growth in China pp 37–79Cite as

Pollution Meets Efficiency: Multi-equation Modelling of Generation of Pollution and Related Efficiency Measures

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Abstract

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 modify 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.

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Notes

  1. 1.

    Xun Zhou studies environmental productivity growth in consumer durables in this volume.

  2. 2.

    As observed in Ayres and Kneese (1969, p. 283) abatement does not “destroy residuals but only alter their form”. Following Russell and Spofford (1972), the concept of “modification” should be used instead of waste treatment or purification to underline the conservation of mass. The mass of residuals does not physically disappear by waste treatment or purification.

  3. 3.

    The materials balance is quite seldom mentioned in papers published in operational research journals or papers written by researchers from that field. In a recent survey article (Sueyoshi et al. 2017) based on 693 papers using data envelopment analysis within energy and environment, materials balance is never mentioned once.

  4. 4.

    This type of model but without reference to Frisch (1965) was used in Mäler (1974). His book was written when the author was a visiting scholar at Resources for the Future (RFF) invited for a year by Allen V. Kneese.

  5. 5.

    This model applied with explicit reference to Frisch (1965) to production of both desirable and undesirable outputs, was, to the best of our knowledge, first used in Førsund (1972, 1973) and developed further in Førsund (1998, 2009, 2018).

  6. 6.

    A production possibility set was also introduced using the transformation relation such that the value of the transformation function is zero for efficient utilisation of resources and less than zero for inefficient operations, but no inefficiency issues were discussed. The solutions to the optimisation problems were based on the production of desirable outputs being efficient.

  7. 7.

    Chinese researchers have written several such papers, see e.g. Wang et al. (2012, 2017a, b); Xie et al. (2017); Zhao et al. (2015) for recent applications to Chinese data.

  8. 8.

    In Førsund (2017), it is argued that desirable and undesirable outputs must be measured in the same unit in order to perform a trade-off, cf. point (d) in Sect. 2.1. This can be done by introducing a damage function, as also used in Førsund (2017, 2018).

  9. 9.

    Førsund (1998) was the first to criticise both the assumption of bads as inputs and the weak disposability assumption. However, the first submission to a journal of the paper was rejected, and an improved version Førsund (2009) published first 11 years later.

  10. 10.

    However, problems with translation properties are pointed out in Aparicio et al. (2016), and infeasibility problems pointed out in Arabi et al. (2015).

  11. 11.

    The text below builds on Førsund (2017), Chapter 8.5.

  12. 12.

    The name is meant to point to the production of both desirable goods and residuals. However, the name is not according to the classical economist, calling by-products commercial outputs, but with less value than other goods. Their word for residuals was waste, see next Sect. 3.1.

  13. 13.

    Dakpo et al. (2016) review weak disposability models and the by-production model. Hampf (2018) reviews single equation models only, and has several critical remarks to the typical Färe et al. models of desirable and undesirable outputs.

  14. 14.

    Førsund (2018), Murty and Russell (2018) and Hampf (2018) are forthcoming in a special issue ‘Good modelling of bad outputs’ of Empirical Economics, see Kumbhakar and Malikov (2018).

  15. 15.

    To say that inputs and outputs can be disposed of by throwing them away (Shephard 1970, p. 14) is not in accordance with economic use of inputs and outputs.

  16. 16.

    According to Kurz (1986, p. 25), Mangoldt’s definition is about the same as the one of Frisch: “pure joint production (or joint production in the technical sense) and what may be called competing, alternative or rival (Edgeworth) production which derives from the fact that a firm’s (given) productive equipment may be used for several purposes.”

  17. 17.

    Using non-jointness when defining joint production seems a little awkward; sounding almost like a contradiction. Nadiri (1987, p. 1028) claims that absence of non-jointness is a crucial test of joint production, in spite of including non-jointness as part of the definition of joint production: “Joint production includes two cases: (1) when there are multiple products, each produced under separate production processes—i.e. the production function is non-joint […]”. He uses the term “intrinsic jointness” when there is jointness in a technical sense.

  18. 18.

    There are unfortunately few references to Frisch (1965) about joint production, neither Chambers (1988); Kohli (1983), nor Nadiri (1987) refer to Frisch. It seems appropriate to make his take on joint production better known.

  19. 19.

    In a very readable book, Whitcomb (1972) discusses the connection between externalities and joint production. He refers both to Frisch (1965) and specifies a variant of the system (4), and to Ayres and Kneese (1969), but does not use the materials balance explicitly in his analysis.

  20. 20.

    Kohli (1983) introduces this case in his Definition 4 on p. 213 and calls it “non-joint in input prices”. This seems a little peculiar name since there are no input prices appearing in the definition (however, duality results use shadow prices). He has no reference to Frisch (1965) that introduced this type of relation decades before.

  21. 21.

    Chambers (1988, p. 289 ) calls the factorially determined functions for generalised fixed coefficients technologies.

  22. 22.

    The assortment property is not discussed in Baumol and Oates (1988, Chap. 4). The materials balance principle is not mentioned in the book.

  23. 23.

    Leontief type models with fixed coefficients are not considered.

  24. 24.

    The Frisch model of factorially determined multi-output equations is not the only model having the sufficient separability properties.

  25. 25.

    One or more service inputs may also be essential, but the point is that residuals are in general an unavoidable feature using material inputs in production. Although \( y = f(x_{M} ,0) = 0 \) we may have \( z = g(x_{M} ,0) > 0 \); e.g. as in a fully automated thermal electricity-generating plant running in a spinning mode (the energy stored by spinning is then not considered an output).

  26. 26.

    In Murty and Russell (2018, p. 15) x M is called jointly essential with z, it is rather obvious that z cannot be zero for x M  > 0, however, Rødseth (2017a) covers the possibilities with the concepts output- and input essentiality.

  27. 27.

    Cf. the famous chocolate production example in Frisch (1935), discussed in Førsund (1999), of substitution between labour and cocoa fat due to more intensive recycling of rejects not filling the forms the more labour and less cocoa fat that are employed producing the same amount of chocolate.

  28. 28.

    Continuing the chocolate example in footnote 27: when so much labour is employed so that all the defect chocolates re-circulated to the production cannot be increased any more employing more labour, based on a given mass of raw material with a minimum of cocoa fat for the required taste.

  29. 29.

    A practical use of the materials balance is the estimation of emission coefficients, e.g. when coal is used in thermal electricity generation, assuming a specific physical composition of coal and optimal running of the process. Then, because the complete contents of coal end up as residuals, knowledge of the combustion process allows the emission coefficients concerning the substances actually discharged to the external environment to be calculated.

  30. 30.

    This trade-off should not be confused with a correlation between y and z depending on indirect effects. Increasing x M in (11) will lead to increases in both y and z, thus we have a positive correlation, increasing x S will increase y but decrease z (if y contains materials), and thus we have a negative correlation.

  31. 31.

    Modification and recycling of residuals using factorially determined multioutput production functions was introduced already in Førsund (1973).

  32. 32.

    Hampf (2014) has a similar specification of the abatement function with primary residuals as input together with a stage-specific amount of non-polluting abatement inputs (same as similar inputs in the production of the good output in stage 1), a shared input of the two stages, and part of the output from stage 1 as an input. The abated amount is the single output, and the secondary residual emitted to the environment is residually calculated as in Førsund (2009, 2018).

  33. 33.

    The abatement stage in Färe et al. (2013, p. 112) does not, somewhat awkwardly, show the use of the abated amount explicitly, neither in the definition of their production possibility set (16) nor in their model Eqs. (18), (19) and (20). In Pethig (2006, p. 189) the primary residual seems to be the output of abatement.

  34. 34.

    See Zhao et al. (2015), Xie et al. (2017) for studies of types of command-and-control and market-based environmental regulation in China.

  35. 35.

    In the case of the presence of embodied technology or vintage capital, a distinction should be made between efficient utilisation of the mix of existing technologies and the most modern technology (Førsund 2010).

  36. 36.

    In Färe et al. (2013, p. 110) it is stated: “which [without a restriction] as pointed out by Førsund (2009) would give us a […] nonsensical result that zero bads can be achieved at no costs […]”.

  37. 37.

    Notice that using input-output type of models does not support the assumption of weak disposability, as is made clear in Fig. 3; the input-output assumption means that there is only a single ratio between the good and the bad, not many as illustrated by the two other trade-off curves. However, notice that the Leontief assumption is valid for the point \( {\bar{\text{u}}} \) only. Furthermore, weak disposability is not a case of Frisch (1965) output couplings as in Eq. (6).

  38. 38.

    Note that Shephard (1970, p. 187) was aware of the fact that production relations need not be of a single-equation type: “It is useful to reiterate at this point that the foregoing assumptions for the production correspondence do not exclude the technology being composed of several processes (or sub-technologies) which are to be jointly planned, as well as situations where joint outputs are inherently involved.”

  39. 39.

    Färe et al. (2008, p. 561) state: […] “disposal of bad outputs is costly—at the margin, it requires diversion of inputs to ‘clean up’ bad outputs” […].

  40. 40.

    A peculiarity with the trade-off in Färe et al. (2013) is that the trade-off occurs with the output for final consumption and the secondary pollutants from the abatement stage, and not between the total output of the good (electricity) and the generation of pollutants in the production stage. However, it is the last trade-off that is the functional trade-off that goes against the materials balance principle in the single-equation model of the production stage.

  41. 41.

    A material balance restriction is mentioned, but not implemented in the empirical model. Weak disposability is assumed.

  42. 42.

    This is also the message in Murty and Russell (2018, p. 18) stating: […] “the complex real-world trade-offs among inputs and outputs in these technologies cannot be captured by a single functional relation. For example, it is impossible for a single function to capture, simultaneously, the positive relations between emissions and emission-causing inputs and the positive relations between emissions and intended outputs.”

  43. 43.

    This apt expression is due to Barnum et al. (2017).

  44. 44.

    Dakpo et al. (2016, p. 356) argue that all the different models introduced should be estimated for comparison. As mentioned previously this is also the approach in Hampf (2018). However, in light of the risk of estimation a ‘false model’, one cannot identify the “best” model in such a way. The only way is to choose the theoretically best model.

  45. 45.

    The multi-equation model in Serra et al. (2014) is based on the development in Førsund (2008) (an improved version of this working paper is Førsund 2009) and Murty et al. (2012). Both polluting and non-polluting inputs are specified to produce residuals emitted to the environment [see their Eq. (3)], i.e. no abatement is taking place.

  46. 46.

    Hampf (2014) also solves separate optimisation problems for the production stage and the abatement stage, but this is done by minimising the weighted emissions in the two stages.

  47. 47.

    Hampf and Rødseth (2015) find that most of the efficiency differences in U.S. power plants measured by electricity generation using coal can be explained by the age of plants.

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Acknowledgment

The chapter is based on a presentation at the 2016 Asia-Pacific Productivity Conference, Nankai University, Tianjin, China, 7–10 July, and building largely on the work in progress for Førsund (2017, 2018). I am indebted to Benjamin Hampf, Robert Russell, Kenneth Løvold Rødseth, Victor V. Podinovski and an anonymous reviewer for challenging and constructive comments improving the chapter.

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  1. Department of Economics, University of Oslo, Oslo, Norway

    Finn R. Førsund

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  1. Finn R. Førsund
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Correspondence to Finn R. Førsund .

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  1. College of Economic and Social Development, Nankai University, Tianjin, China

    Ruizhi Pang

  2. College of Economic and Social Development, Nankai University, Tianjin, China

    Xuejie Bai

  3. School of Economics, University of Queensland, Brisbane, Queensland, Australia

    Knox Lovell

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Førsund, F.R. (2018). Pollution Meets Efficiency: Multi-equation Modelling of Generation of Pollution and Related Efficiency Measures. In: Pang, R., Bai, X., Lovell, K. (eds) Energy, Environment and Transitional Green Growth in China. Springer, Singapore. https://doi.org/10.1007/978-981-10-7919-1_3

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