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

Thus the revenue function can be derived from the output
distance function by maximising revenue over output quantities and the
output distance function can be derived by maximising the revenue
function over output prices. This duality between the output distance
function and the revenue function can be used to derive the shadow
prices of the outputs. These are relative output shadow prices and in
order to obtain absolute shadow prices additional information regarding
the revenue is required (Fare et., al. 1993). In order to derive the shadow
prices of outputs we assume that the revenue and distance functions are
differentiable. We follow the methodology used by Fare et., al. (1993)
and write the Lagrangian function as

(vi)KK maxΛ=ry + λ (D₀ (x, y) -1)

The first order conditions with respect to outputs are

( viⁱ )......r = ^λv yD o( x^, y )

where r and the gradient vector V_yD₀(x,y) are of dimension (MxI)
and λ is a scalar. Following Fare et., al. (1993) and with a distance
function which is homogeneous of degree +1 in output y it can be shown
that

(viii) KK -λ = Λ

Thus at the optimum, we have

(ix) KK -λ = Λ = R(x, r)

and equation (vii) can be written as

( ^x )......^r = R ( ^{x, r} ) * ^v y^D o( ^x, У )

In order to establish the relation between the gradient vector
V_yD₀ (x, y) and the shadow prices, we make use of the Shephard’s
duality theorem (v), namely,

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