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A hedonic-output-index-based approach to modeling polluting technologies

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Despite some recent criticisms, the conventional radial distance function, which treats undesirable by-products as either frontier shifters or inputs, remains a popular go-to formulation of polluting production processes among practitioners. This unfading popularity is arguably driven by the ability of radial distance functions, unlike alternative directional distance functions, to allow for unit-free multiplicative changes in arguments as well as, by implicitly postulating the radial direction, to free researchers from the dilemma of having to explicitly choose the directional vector. In this paper, we offer a generalization of the standard radial distance function to polluting technologies that can accommodate undesirable by-products in a more economically meaningful way. Specifically, we propose modeling undesirable outputs via a hedonic output index, which is meant to ensure that pollutants are treated as outputs, as opposed to inputs or theoretically unregulated frontier shifters, while also recognizing their undesirable nature. By using a radial input distance function generalized to encompass an (unobservable) hedonic output index of desirable and undesirable outputs, we are able to meaningfully describe relationships between different products (including the complementarity of desirable and undesirable outputs) within producible output sets as well as to represent technically feasible polluting production possibilities given inputs. An empirical application of our methodology to the case of Dutch dairy farms in 2001–2009 demonstrates the complexity of interactions between outputs, thereby attesting to the value of more elaborate representations of production possibilities.

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  1. Note that the constant term in the translog specification of the hedonic output index is normalized to zero for identification purposes.

  2. Recall that the linear homogeneity properties are already imposed by construction.

  3. Quasi-maximum likelihood is another feasible alternative.

  4. Note that no “violations” are computed for the same elasticity in the case of Model (II) since this model does not theoretically regulate the sign of monotonicity of the IDF with respect to undesirable by-products.

  5. Following Malikov et al. (2015b), the concept of returns to scale is defined over desirable outputs only.

  6. Standard errors for the returns to scale estimates are constructed using the delta method. In turn, the standard errors for the IDF parameter estimates are computed using the robust outer-product sandwich variance–covariance matrix.

  7. It is also noteworthy that the two papers use different formulations of the production technology. Brümmer et al. (2002) estimate a conventional input distance function, whereas Emvalomatis et al. (2011) specify an output distance function.

  8. It might appear that one cannot use the same methodology to compute the shadow price of b for Model (III) because it treats b as an input. However, the formula for r even when b is an input stays unchanged if one replaces the revenue maximization framework with the profit maximization in which b is treated as one of the inputs.

  9. To be able to compare our shadow price estimates with those reported by Reinhard et al. (1999), we convert their average shadow price estimate measured in 1991 guilders into euros of 2005.

  10. Morishima elasticity computed using the IDF estimates from the other two models will lack a proper economic interpretation.

  11. We employ an axis-aligned bivariate Gaussian kernel, evaluated on a square grid using the normal reference bandwidth.


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Correspondence to Subal C. Kumbhakar.

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The data used in the present paper come from the Dutch FADN system as collected by the Dutch Agricultural Economics Research Institute (LEI). The Centre of Economic Information (CEI) has provided access to these data. The reported results are and remain entirely the responsibility of the authors; they neither represent the views of LEI/IEC nor do they constitute official statistics. Bokusheva also acknowledges financial support by the Swiss National Science Foundation (Research Grant No. 100014_128967).

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Malikov, E., Bokusheva, R. & Kumbhakar, S.C. A hedonic-output-index-based approach to modeling polluting technologies. Empir Econ 54, 287–308 (2018).

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