Estimation of marginal abatement costs for undesirable outputs in India's power generation sector: An output distance function approach.

II. Theoretical Model

The conventional production function is defined as the maximum
output that can be produced from a given vector of inputs. The distance
function generalises this concept to a multi-output case and describes
how far an output vector is from the boundary of the representative
output set. We can define the output distance function in terms of the
output set P(x) . Suppose that a producer employs the vector of inputs
x ∈ R₊^N to produce the vector of outputs y ∈ R₊^M , where R₊^N , R₊^M are
non-negative N and M dimensional Euclidean spaces, respectively. The
plant technology captures the relationship between the inputs and
outputs and is described by the output set P(x) . The output setP(x)
denotes all output vectors that are technically feasible for any given input
vector x, i.e.,

(i) KK P(x) = {y ∈R₊^M :x can produce y}

The output set is assumed to satisfy certain axioms, the details of which
can be seen in Fare (1988). The output distance function is defined on
the output set P(x) as

(ii)......D0 (x, y) = min{θ > 0:(y /θ ) ∈ P(x)}∀x∈R+^v

θ

The above equation measures the largest radial expansion of the output
vector y, for a given input vector x, that is consistent with y belonging to
P(x) . The value of the output distance function must be less than or
equal to one for any feasible output. The axioms regarding the output set
P(x) impose a set of properties¹ on the output distance function which
are as follows:

1. D₀ (0, y) = + ∞ for y ≥ 0, i.e., there is no free lunch. To produce
outputs one requires inputs.

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