3 The paper unfolds as follows The following section will introduce the

# 3 the paper unfolds as follows the following section

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The paper unfolds as follows. The following section will introduce the methodology.Section 3 is allocated to the presentation of the data source and discussion of results.Finally, section 4 concludes.2. MethodologyIn this section we describe the environmental index adopted in this paper3. In short, theindex is defined as the ratio of a good output quantity index and a quantity index of bador undesirable outputs. Each of the two indexes are based on distance functions, verymuch like the Malmquist (1953) index, but rather than scaling the full output vector, wescale good and bad outputs separately. Thus our index is developed using "sub-vector"distance functions. To describe the environmental performance index some notation is needed. Assume that avector of inputs NNRxxx),.....,(1produces a vector MMRyyy),....,(1of goodoutput and at the same time produces a vector JJRbbb),....,(1of bad outputs, thenwe define the production technology as )},(producecan :),,{(byxbyxT.This technology producing both good and bad outputs is assumed to satisfy the followingtwo conditions.Weak disposability of outputs: TbyxTbyx),,(,10and),,(ifNulljointness: 0then 0and),,(ifybTbyx.Weak disposability models the following situation: reduction in outputs ),(byarefeasible, provided that they are proportional. This means proportional contraction ofoutputs can be made, but it may not be possible to reduce any single output by itself. Inparticular it may not be possible to freely dispose of a bad output.4
The nulljointness condition tells us that for an input output vector ),,(byxto befeasible with no bad output )(bproduced, it is necessary that no good output )(yisproduced. Put differently if good output is produced, some bad output is also produced.In addition to the above two properties on the technology T, we assume that it meetsstandard properties like closedness and convexity. See Färe and Primont (1995) fordetails. An output set )(xPthat satisfies weak disposability and nulljointness propertiesis illustrated in Figure 1. We note that for each ),(byin )(xP, proportionalcontractions of any feasible output pair are feasible. Also the good and bad outputs arenulljoint. If 0bthen 0ywhenever ),(byis in )(xP.Please insert Figure 1 hereTo formulize the good output quantity index, we define a subvector output distancefunction on the good outputs as}),/,(:inf{),,(TbyxbyxDy.This distance function expands good outputs as much as is feasible, while keeping inputsand bad outputs fixed. Note that it is homogeneous of degree +1 in y. Let 0xand 0bbe our given inputs and bad outputs, then the good output index compares two outputvectors kyand ly. This is done by taking the ratio of two distance functions, and hence,the good index is:),,(),,(),,,(000000byxDbyxDyybxQlykylky.

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• Spring '14
• DanielKevles
• Environmental Economics, Per capita income, Random effects model, environmental performance, Environmental Performance Index