Nonlinear Production, Abatement, Pollution and Materials Balance Reconsidered

=Xr=0,
r_c

X =X =0 and X >0.
ra1        ra2                       rc

To derive (22) from (21), we first consider (21f) and (21g) to obtain — = —----

λfλf

λ

= 1 + — X, . Next we combine this equation with (21a) to turn (21b) into
λf   ^c

λx
λf

^Uxⁱ

∑ⁱuⁱ_y

Uⁱ

1+X ∑^Ux
rc    i_Ui

^y
since λ_x∣λf > 0 in an interior solution. We proceed by infer-
this equation into (24).

Clearly, (22) characterizes the efficient level of pollution generated jointly via the emission of
all residuals. MD_x is the damage from a marginal increase in pollution evaluated by the con-
sumers' aggregate marginal willingness-to-pay in terms of the consumer good for avoiding
that increase. The right side of (22) represents the marginal benefit of pollution that comes in
form of an increase in the consumer good made possible by a small increase in pollution
through stepping up emissions. Equivalently, one can interpret MD_x as the marginal benefit
and Y_eQ∕X_e as the marginal cost of reducingpollution. To further interpret the term Y_eQ∕X_e
observe that Y_e is the amount of consumer goods that cannot be produced anymore when the
emission of production residuals is reduced by a small amount. It is known as marginal
abatement costs of production residuals (in terms of the consumer good) in models where no
residuals other than production residuals are considered. For convenience, we will refer to Y_e
as conventional marginal abatement costs of production residuals.

If we multiply both sides of (22) by X_e we obtain MD_x ■ X_e = Y_e ■ Q. The left side of this
equation represents the marginal benefit of reducing the emission of production residuals and

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