(a good output) and the quantity of SO2 emissions (a bad output which locally has a
direct negative effect on health and welfare and regionally can lead to acid rain). Three
inputs are applied to produce these outputs: capital, labor, and energy. This application is
particularly relevant since allowable SO2 emissions from electric utilities have been reduced
dramatically over the last decade and since electricity is currently in short supply in the
State of California, where State Implementation Plans are very strict. Title IV of the
1990 Clean Air Act Amendments reduced emissions of SO2 from U.S. coal-burning electric
utilities from about 19 million tons in 1980 to 8.95 million tons by the year 2000. The
increased reduction of SO2 emissions over time has likely had an important impact on the
levels of technical efficiency for these utilities. Proper crediting for reduction of this bad is
essential to obtain unbiased estimates of efficiency levels. It also can provide insights into
what the tradeoff has been between emissions and output.
4.1 The Model
Let x be a vector of inputs x = (x1 , . . . , xN) ∈ R+N and let y be a vector of good out-
puts denoted by y = (y1, . . . , yM) ∈ R+M. Disregarding bads, one can write the production
technology, S(x, y, t), as
S(x, y, t) = {(x, y) : x can produce y at time t}, (4.1)
where t = 1, . . . , T is time.
This application, however, has a bad output (air pollution) that must be accounted
for to accurately measure the technical efficiency of the various utilities. Ignoring the bad
would allow a firm to look more efficient by ignoring the environment, while a firm that