Nonlinear Production, Abatement, Pollution and Materials Balance Reconsidered



17


MDx ■ Xe {>} Ye

xee


if


X >0 and/or X >0 and X =0,
ra1ra2rc



=Xr=0,
rc


(23)


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

iuiy

(24)


Ui

1+XUx
rc    iUi

f U U1 ^ r
In (24), 1 + Xrc ∑Uχ ]0,1[
к        Uy J


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

λx
ring x

λf


Ye

Xe +Σ jXrajRe


from (21c), (21h) and (21k). (22) is then obtained by inserting


Clearly, (22) characterizes the efficient level of pollution generated jointly via the emission of
all residuals. MDx 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
MDx as the marginal benefit
and
YeQ∕Xe as the marginal cost of reducingpollution. To further interpret the term YeQ∕Xe
observe that Ye 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
Ye
as conventional marginal abatement costs of production residuals.

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



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