Nonlinear Production, Abatement, Pollution and Materials Balance Reconsidered

21

Maintaining the assumption X_e > 0 (which implies t_e^A > 0 owing to (29)) we now wish to
juxtapose alternative scenarios where one, two or all of the (efficient!) tax rates t_a^A₁,t_a^A₂ and
t_c^A may be zero. In view of Y_e>0,R_e¹ <0 and R_e² < 0 , it is straightforward that (29) and (30)
imply (28). (28) complements (24), in fact, but (28) was derived via (30) rather than via (23)
and thus offers insights in the determinants of the efficient emissions tax rate t_e .

Suppose first, pollution is caused by the emission of production residuals only
(X_ra1= X_ra2= X_rc =0). Then t_e= Y_e is optimal due to (28), i. e. the tax rate t_e is required to
be set equal to the conventional marginal costs of abating production residuals. When com-
bined with (29) t_e= Y_e yields

_Ui_X

-∑ U_ = t = Y                                         (31)

ii ^ee

U_y

which constitutes the conventional Pigouvian tax rule. Yet in view of possible deviations
from that rule, as described in (29) and (31), (31) is not a trivial result because we specified
conditions under which the conventional Pigouvian tax rule is valid.

Suppose next that __r > 0 and/or __r > 0 but __r = 0 . In this case the social marginal
ra1                            ra2                      rc

abatement costs of production residuals are Y_e-∑_jt_a^A_jR_e^j> Y_e. The extra marginal costs,
∑ _j t_aj R_e^j , accrue because an increase in the abatement of production residuals inadvertently
raises the generation of environmentally harmful abatement residuals. The result t_e^A > Y_e is
worth noting: It is now optimal to abate less production residuals than under the conventional
Pigouvian tax rule. Conversely, if abatement residuals don't degrade the environment but
post-consumption residuals do (__ra1= __ra2=0 and __rc>0), then the social marginal abate-
ment costs of production residuals are Y_e-t_c^AY_e< Y_e. The extra marginal cost savings, t_c^AY_e,
occur because an increase in the abatement of production residuals inadvertently reduces the
amount of post-consumption residuals and consumer goods. Since post-consumption residuals
are (now) pollutants by assumption, curbing r_c reduces that pollution which is, in turn, a
beneficial external effect of abating production residuals. With t_e^A < Y_e it is now optimal to
abate more production residuals than under the conventional Pigouvian tax rule.

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