MULTIPLE COMPARISONS WITH THE BEST: BAYESIAN PRECISION MEASURES OF EFFICIENCY RANKINGS

spent effort on mitigating its pollution would be unrewarded and measured as relatively
inefficient.

Symmetric treatment of bads (denoted by a vector b) and goods using an input
distance function is legitimate and can be specified as

Di([y^,b],x^,t) = sup{^λ : ([y^,b],^χA) ∈ S(x^, b^,y^,t)}.                 (4.2)

λ

Here the goods and bads are held constant and inputs are proportionally scaled downward
to their minimum required level. Since the input distance function in (4.2) is dual to the
cost function, we can write

Ci([y, b], p, t) = min{px : Di ([y, b], x, t) ≥ 1},                    (4.3)

x

where p = (p1, . . . , pN) ∈ R₊^N is a vector of input prices and C ([y, b], p, t) is a unit
cost function if costs are minimized. This equation implies that unless inputs are used in
their cost-minimizing proportions, the input distance measure will be greater than one.
Formulating the associated Lagrangian and taking the first-order conditions, Fare and
Primont (1995) show that the shadow value for each input is given by

p = C([y^, b], P,^t)VχDi([y, b], x,^t).                             (4.4)

Equivalently, the bads can be treated as exogenous shifters of the technology set,
similar to a time trend or state of technology variable. The intuition is that conditional
on the level of the bad, efficiency measures over the desirable outputs and inputs are well-
defined and behave as expected. Yet, ignoring the bads would lead to biased results since
firms would not receive credit for input use that is directed at reducing output levels of
the bad. Treating the level of the bad as a shifter of the technology set allows firms to be
credited (penalized) for reducing (increasing) the level of bad that they produce.

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