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

To emphasize this point, equation (4.2) can be written as

Di (y,x,t∣b) = sup{λ : (x∕λ,y∣b) ∈ S(x,y,t∣b)}.                   (4.5)

λ

The appropriate monotonicity condition for the bad in the context of the input dis-
tance function can be derived as follows. Assuming a single bad, we compute the partial
total differential of equation (4.5) evaluated on the frontier at a fixed time [implying
D_i(y, x, t|b) = 1 and dt = 0] to obtain

7 ∩ ∖ - ^dDi ₁ . ∖ ^dDi ₁ . ^dDi ,, ∩
dDi = > _y ɑ—dym + > _j d—dxn + -^~db = 0.
∂              y_m ∂ x_n∂ b

Using the properties that the input distance function is monotonically nondecreasing
in inputs (^d∂χDⁱ ≥ 0) and monotonically nonincreasing in outputs (∂D-ⁱ ≤ 0), and setting
dy_m = 0, ∀m, in order to keep the firm on the input distance frontier, we obtain

As in Pittman (1983), with constant desirable output and technology, bads can only be
reduced through increased input usage. This implies that ddn ≤ 0, which combined with
the nonnegativity property for inputs, ddDDi ≥ 0, yields

∂Di

ɪ ≥ 0.
∂b

As a flexible approximation to the true distance function in (4.5) , we adopt the
translog functional form. Thus, the empirical model for firm f = 1, . . . , F in period

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