Nonlinear Production, Abatement, Pollution and Materials Balance Reconsidered

16

as the pollution that results from releasing the residuals e, r_a1, r_a2 and r_c into the environment.
X is assumed to be non-decreasing and convex. If X_v = 0 for v= r_a1,r_a2,or r_c the level of the
residual under consideration is small enough to be fully neutralized by nature's assimilative
capacity. Environmental degradation as described by x negatively affects the consumers' util-
ity

u_i = U (x, y_j)    i = 1, ..., n.                                                             (18)

-+

The general equilibrium model is completed by introducing the standard resource constraints

y ≥∑_iyi , £ ≥ £ and m ≥ m,
where £ and m are the economy's fixed factor endowments. To characterize a Pareto effi-
cient allocation of the economy (15) - (19) we solve the Lagrangean

Σ i^αU^t ( ^x, y_i ) ^{+ λ}f ^[ Y ( ^{e, £,} m )^- У ] ^{+ λ}x [ ^{x -} X ( ^e, ra 1 ^,ra 2^, rc )]^{+ λ}1 ^[ ra 1 ^- R 1 (^{e, £,} m )] ^+
+λ2 [ ra 2 ^- R 2 ( e^{, £,} m )] ^{+ λ}y ( У ^- Σ _i У г )^{+ λ}c ( rc ^- У ) ^{+ λ}l ( ^{- £} ) ^{+ λ}m ( m ^- m ) ^{,          (20)}

where α_i are arbitrary positive numbers for i = 1, ..., n. Under the assumption that an inte-
rior¹¹ solution exists the first-order conditions are

Proposition 4: The efficient allocation of the economy (15) - (19) is characterized by

Y                     Uⁱ

= —— ■ Q with MD_r : = -∑ —- and Q :=

X_e              x        Ui_y

As a consequence,

¹¹ In the present context, an interior solution implies that (e, £, m) is in the interior of the domain D of function
Y. This doesn't only require all variables to be bounded away from zero but also that the inequality
e ≤ m - F (£, m) is not binding. For the sake of completeness one would have had to consider this inequality as
an additional constraint in (20). But with the strict inequality sign in the solution the associated Lagrange multi-
plier would be zero and therefore need not be introduced in the first place.

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