16
as the pollution that results from releasing the residuals e, ra1, ra2 and rc into the environment.
X is assumed to be non-decreasing and convex. If Xv = 0 for v= ra1,ra2,or rc 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
ui = U (x, yj) 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
(19)
Σ iαUt ( x, yi ) + λf [ Y ( e, £, m )- У ] + λx [ x - X ( e, ra 1 ,ra 2, rc )]+ λ1 [ ra 1 - R 1 (e, £, m )] +
+λ2 [ ra 2 - R 2 ( e, £, m )] + λy ( У - Σ i У г )+ λc ( rc - У ) + λl ( - £ ) + λm ( m - m ) , (20)
where αi are arbitrary positive numbers for i = 1, ..., n. Under the assumption that an inte-
rior11 solution exists the first-order conditions are
|
λy=αiUiy i = 1, ., n |
(21a) |
λy= λf+ λc |
(21f) |
|
λ-=-∑iαiU-i |
(21b) |
λc= λ-Xrc |
(21g) |
|
λfYe=λ-Xe+∑jλjRej |
(21c) |
λ1 = λ X 1 - ra1 |
(21h) |
|
λfYi = λ +∑ j λR |
(21d) |
λ2 = λ X 2 - ra2 |
(21k) |
|
λfYm=λm+∑jλjRmj |
(21e) |
Proposition 4: The efficient allocation of the economy (15) - (19) is characterized by
MDx
Y Ui
(22)
= —— ■ Q with MDr : = -∑ —- and Q :=
Xe x Uiy
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.