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

( xix) KK max ∑ [ln D₀ (x^k , y^k )-ln1]
k=1

where k = (1, 2,......, K) indexes individual observation. lnD₀ (x, y)

has an explicit functional form as given in equation (xvi). We assume that
the first i outputs are desirable while the remaining (M - i) outputs are
undesirable or bad outputs. Our objective function minimises the sum of
deviations of individual observations from the frontier of the technology.
We know that the distance function takes a value less than equal to
unity, therefore the natural logarithm of it, i.e., lnD₀(x^k , y^k) will be less
than, or equal to zero and the expression [ln D₀ (x^k , y^k)-ln1] , which
denotes the deviation from the frontier for observation k will be less than
or equal to zero.

Our objective is to maximise the expression in equation (xix)
subject to the following constraints:

(xx)KK lnD₀(x^k, y^k)≤0, k=1,......,K

This constraint restricts the individual observations to be either on or
below the frontier of technology i.e., there are no outputs outside the
frontier of technology, given the set of inputs.

Desirable outputs are assumed to be strongly disposable, which
implies that the output distance function should be increasing in
desirable outputs. The strong disposability condition can be represented
by the following inequality:

(xxi)......^d lnD⁰(x.,y ) ≥ 0,  m = 1,......,i;  k = 1,......,K

∂ ln y^k_m ’        ’    ”        ’    ’

The constraint above ensures that the shadow prices of the desirable
outputs are non-negative. In addition it is assumed that undesirable
outputs are weakly disposable. This weak disposability is always
satisfied for the output distance function specified as the translog form
when linear homogeneity condition represented by equation (xxii) and
the symmetry conditions represented by equation (xxiii) are being
imposed.

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