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
Di(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.
∂ ym ∂ xn∂ b
(4.6)
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
dym = 0, ∀m, in order to keep the firm on the input distance frontier, we obtain
∂Di
∂b
∂Di dxn
∂xn db
(4.7)
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
(4.8)
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
11
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