Nonlinear Production, Abatement, Pollution and Materials Balance Reconsidered

19

from that conventional Pigouvian tax rule required in a model containing (15) and (17) and
when X_ra1,X_ra2 and/or X_rc are positive.

To address this issue we need to look at the agents' optimization problems as implicitly de-
scribed by the Lagrangeans¹³

U (^Х,Уг ) ⁺ μ∙[PÂ ⁺ Pmmi ^{+ θ}i (g ⁺ b)- РуУ ^] and                            ⁽²⁵⁾

{РуУ -(Pt ⁺ tt ) ^t -(Pm ⁺ tm ) m - ta 1 ra 1 ^- ta2ra2 ^- tcrc - tee} ⁺ μc (rc - У ) +

⁺μf [Y (^{e, t,} m)- У] ⁺ μ [r_ai - R¹ (e, ^t, m)^] + μ2 [r_a2 - R² (e, ^t, m)].                ⁽²⁶⁾

With regard to post-consumption residuals, implicit in (25) and (26) is the so-called take-back
rule as modeled, e. g., in Eichner and Pethig (2000). Each consumer i purchases the amount
y_i of consumer goods, and after consumption she returns the post-consumption residuals to
the producers. By institutional design, producers are responsible for the (orderly) deposition
of these residuals, and they may be therefore charged an emission fee, t_c , if the post-
consumption residuals cause pollution after having been emitted. This institutional arrange-
ment explains why post-consumption residuals don't enter the consumers' optimization calcu-
lus (25),¹⁴ but rather are part of the (aggregate) firm's profit maximization calculus.

The (aggregate) firm employs the technology (15) and is charged input taxes, t_t and t_m, as
well as emission taxes, t_e,t_a1,t_a2 and t_c . Some of these tax rates may be zero, of course. The
first-order conditions associated to an (interior) solution to (25) and (26) are conveniently
summarized in

μ∙Py = Uy^,                (²⁷a)         ( Py - tc ) Y = P t ⁺ tt ⁺∑ jjt ^,     (^27c)

( Py - ^tc ) Ye = ^te ⁺Σ JtaJRe ^{, (27b)}         ( Py - tc ) Ym = Pm ⁺ tm ⁺Σ jtaR ■    ⁽²⁷d)

Proposition 4:

¹³ For v = t, m, consumer i's factor endowment is v_i with ∑ ~_i = V. θ. is her share in the firm's profit, g, and
the government's budget surplus, b. The consumers' profit and surplus shares satisfy ∑ _j θ_j = 1 .

¹⁴ One can easily account for that alternative by modifying (25) and (26) in the following way: In (25), add the
term " -t_cr_ci " to the second expression in brackets and add the Lagrange constraint"λ_cir_ci(r_ci-y_i)" at the end.
In (26), delete the term " -t_cr_c " and the Lagrange constraint " μ_c (r_c - y) ". In the absence of institution-specific
costs the conditions for allocative efficiency are the same in both regimes (Eichner and Pethig 2000).

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