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Environmental Policy in Vertical Markets with Downstream Pollution: Taxes Versus Standards

  • Yang-Ming Chang and Manaf Sellak ORCID logo EMAIL logo

Abstract

This paper examines the performance of two environmental regulation policies – emission taxes and absolute standards – in a vertical market where an upstream foreign monopolist sells a specific input to two downstream multiproduct firms that generate pollution in the domestic country. Specifically, we use a three-stage game to analyze and compare the two policies for regulating downstream pollution. In the first stage, the domestic government determines an optimal tariff and sets one of the two instruments (taxes or standards) by maximizing social welfare, in stage two, the upstream foreign monopoly sets its input price, and finally, the downstream domestic firms independently make their output and abatement decisions for profit maximization. We find that total emissions are lower under the absolute standard. Nevertheless, the tax dominates the standard in terms of domestic welfare, consumer surplus, and downstream multiproduct firms’ profits. Thus, the tax equilibrium leads to a win-win-win situation compared to the standard equilibrium. These results show the non-equivalence of emission taxes and absolute standards in regulating downstream pollution. The analyses suggest that a pollution tax is an economically and politically feasible policy.

JEL Classification: Q58; L11; L22

Corresponding author: Manaf Sellak, School of Business, Washburn University, Rm 310M Henderson Learning Center, 1700 SW College Ave., Topeka, KS 66621, USA, E-mail:

We are deeply grateful to Till Requate, Editor in Chief, and two anonymous reviewers for insightful comments and valuable suggestions, which led to substantial revisions in the analytical framework and significant improvements in the paper. The usual disclaimers apply.


  1. Compliance with ethical standards: The authors declare no conflict of interest. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Appendix A

A.1 Welfare Maximization Problem under an Emission Tax Policy

Under an emission tax policy, the government determines optimal rates on tariff and emission tax to maximize social welfare. Making use of Equations (3) and (4) and (6)a)–a(9a), the government solves the following welfare maximization problem:

Max { τ , t } SW T = CS T + π 1 T + π 2 T + GR T ED T

where GR T = τ(q A1 + q B1 + q A2 + q B2) + t[(q A1a A1) + (q A2a A2)], which is the sum of tariff revenues and emission taxes.

To find the solution, we first use the DMP firms’ output levels in (12) to calculate consumer surplus, the firms’ profits in (3), tariff revenue, emission taxes, and the environmental damage in (5). We record the results as follows:

CS T = 4 ( 1 β ) ( α τ ) ( α τ t ) + t 2 ( 3 β + 5 ) 36 ( 1 β ) ( 1 + β ) ,

π 1 T = π 2 T = 4 ( 1 β ) ( α τ ) ( α τ t ) + ( 59 + 3 β 54 β 2 ) t 2 72 ( 1 β ) ( 1 + β ) ,

GR T = ( 1 β ) [ 4 τ ( α τ ) + 2 t ( α 2 τ ) ] t 2 ( 3 + β ) 6 ( 1 β ) ( 1 + β ) ,

ED T = λ e A 1 2 + e A 2 2 = [ 2 ( 1 β ) ( α τ ) t ( 21 + β 18 β 2 ) ] 2 108 ( 1 β ) 2 ( 1 + β ) 2 ,

We then use the welfare function to calculate the FOCs for the government.

SW T t = 303 t 48 α + 66 τ + 42 t β + 46 α β 64 β τ 455 t β 2 36 t β 3 + 162 t β 4 + 44 α β 2 42 α β 3 62 β 2 τ + 60 β 3 τ 54 ( 1 β 2 ) 2 = 0

SW T τ = ( 33 t 8 α + 26 τ + t β + 2 α β 2 β τ 30 t β 2 + 6 α β 2 24 β 2 τ ) 27 ( 1 β ) ( 1 + β ) 2 = 0

Solving the FOCs for the optimal tariff and emission tax yields

τ T = α ( 286 240 β 285 β 2 + 243 β 3 ) 1177 1020 β 1113 β 2 + 972 β 3 > 0 a n d t * = 6 α ( 1 β ) ( 10 9 β ) 1177 1020 β 1113 β 2 + 972 β 3 > 0 .

Plugging τ T and t* back into the relevant equations under the emission tax policy, we have their equilibrium values as summarized in the first lemma (see Appendix A.1).

Lemma 1

In a vertical market where domestic polluting firms produce downstream and rely on an intermediate input imported from an upstream foreign monopolist, imposing an emission tax to regulate downstream pollution leads to the following equilibrium results:

w T = α ( 1433 1203 β 1425 β 2 + 1215 β 3 ) 2 ( 1177 1020 β 1113 β 2 + 972 β 3 ) ,

q A 1 T = q A 2 T = 3 α ( 1 β ) ( 89 81 β ) 2 ( 1177 1020 β 1113 β 2 + 972 β 3 ) ,

q B 1 T = q B 2 T = α ( 307 546 β + 243 β 2 ) 2 ( 1177 1020 β 1113 β 2 + 972 β 3 ) ,

CS T = α 2 ( 82 769 221 823 β + 116 238 β 2 + 161 334 β 3 197 559 β 4 + 59 049 β 5 ) ( 1177 1020 β 1113 β 2 + 972 β 3 ) 2 ,

π 1 T = π 2 T = α 2 88 169 242 343 β + 145 452 β 2 + 142 866 β 3 193 185 β 4 + 59 049 β 5 2 ( 1177 1020 β 1113 β 2 + 972 β 3 ) 2 ,

π U T = 3 α 2 ( 1 + β ) ( 287 528 β + 243 β 2 ) 2 ( 1177 1020 β 1113 β 2 + 972 β 3 ) 2 ,

GR T = 2 α 2 ( 84 692 229 887 β + 128 778 β 2 + 152 748 β 3 195 372 β 4 + 59 049 β 5 ) ( 1177 1020 β 1113 β 2 + 972 β 3 ) 2 ,

ED T = 3 α 2 ( 1 β ) 2 ( 27 29 β ) 2 ( 1177 1020 β 1113 β 2 + 972 β 3 ) 2 ,

MED T = 2 α ( 1 β ) ( 29 27 β ) 1177 1020 β 1113 β 2 + 972 β 3 ,

SW T = α 2 ( 287 528 β + 243 β 2 ) 1177 1020 β 1113 β 2 + 972 β 3 .

A.2 Welfare Maximization Problem under an Absolute Standard Policy

Under an absolute standard, the government solves the following welfare maximization problem:

Max { τ , s } SW S = CS S + π 1 S + π 2 S + GR S ED S

where GR S = τ(q A1 + q A2 + q B1 + q B2), which is the amount of tariff revenues collected by the government. To find the solution, we first use the DMP firms’ output levels in (15) to calculate consumer surplus, the DMP firms’ profits in (4), tariff revenue, and the environmental damage in (5). We note the terms of the welfare functions as follows:

CS S = 1 2 Q A 2 + 2 Q A Q B + Q B 2

where Q A = q A1 + q A2 and Q B = q B1 + q B2 with the output levels of the products by the DMP firms as given (20),

π 1 S + π 2 S = 110 α 33 s 110 τ 60 s β 189 α β + 189 β τ + 81 s β 2 + 81 α β 2 81 β 2 τ 6 ( 11 9 β 2 ) ( 10 9 β ) ,

GR S = 2 τ [ 3 ( 1 β ) s + ( α τ ) ( 10 9 β ) ] 3 ( 11 9 β 2 ) a n d ED S = 1 3 ( 2 s ) 2 .

We then use the welfare function to calculate the FOCs for the government.

SW S τ = CS S τ + π D 1 S + π D 2 S τ + GR S τ ED S τ = 0 ,

SW S s = CS S s + π D 1 S + π D 2 S s + GR S s ED S s = 0 .

Solving the FOCs for the optimal levels of import tariff and emission standard yields

τ S = α ( 6581 12 393 β + 933 β 2 + 9261 β 3 4374 β 4 ) 25 262 46 476 β + 726 β 2 + 38 016 β 3 17 496 β 4

and

s * = 3 α ( 1 β ) ( 10 9 β ) ( 29 27 β ) 2 ( 12 631 23 238 β + 363 β 2 + 19 008 β 3 8748 β 4 ) .

Plugging τ S and s* back into the equations under the absolute standard policy, we have the second lemma:

Lemma 2

The imposition of an emission standard policy to control downstream pollution leads to the following equilibrium results:

w S = α ( 16 052 29 817 β + 1203 β 2 + 23 517 β 3 10 935 β 4 ) 2 ( 12 631 23 238 β + 363 β 2 + 19 008 β 3 8748 β 4 ) ,

q A 1 S = q A 2 S = 3 α ( 1 β ) ( 10 9 β ) ( 89 81 β ) 2 ( 12 631 23 238 β + 363 β 2 + 19 008 β 3 8748 β 4 ) ,

q B 1 S = q B 2 S = α ( 10 9 β ) ( 307 546 β + 243 β 2 ) 2 ( 12 631 23 238 β + 363 β 2 + 19 008 β 3 8748 β 4 ) ,

CS S = α 2 ( 10 9 β ) 2 82 769 221 823 β + 116 238 β 2 + 161 334 β 3 197 559 β 4 + 59 049 β 5 ( 12 631 23 238 β + 363 β 2 + 19 008 β 3 8748 β 4 ) 2 ,

π 1 S = π 2 S = α 2 ( 10 9 β ) 2 93 389 262 341 β + 174 162 β 2 + 124 560 β 3 188 811 β 4 + 59 049 β 5 2 ( 12 631 23 238 β + 363 β 2 + 19 008 β 3 8748 β 4 ) 2 ,

π U S = 3 α 2 ( 10 9 β ) ( 11 9 β 2 ) ( 287 528 β + 243 β 2 ) 2 ( 12 631 23 238 β + 363 β 2 + 19 008 β 3 8748 β 4 ) 2 ,

GR S = α 2 ( 10 9 β ) ( 287 528 β + 243 β 2 ) 6581 12 393 β + 933 β 2 + 9261 β 3 4374 β 4 ( 12 631 23 238 β + 363 β 2 + 19 008 β 3 8748 β 4 ) 2 ,

ED S = 3 α 2 ( 1 β ) 2 ( 10 9 β ) 2 ( 29 27 β ) 2 ( 12 631 23 238 β + 363 β 2 + 19 008 β 3 8748 β 4 ) 2 ,

MED S = 2 α ( 1 β ) ( 10 9 β ) ( 29 27 β ) 12631 8748 β 4 + 19 008 β 3 + 363 β 2 23 238 β ,

SW S = α 2 ( 10 9 β ) ( 287 528 β + 243 β 2 ) 12 631 23 238 β + 363 β 2 + 19 008 β 3 8748 β 4 .

A.3 A Comparison in Equilibrium Outcomes Between Taxes and Standards

A.3.1 Tariff Rates

τ T τ S = α ( 240 β 285 β 2 + 243 β 3 + 286 ) 1020 β 1113 β 2 + 972 β 3 + 1177 α ( 6581 12 393 β + 933 β 2 + 9261 β 3 4374 β 4 ) 25 262 46 476 β + 726 β 2 + 38 016 β 3 17 496 β 4

= 3 α ( 1 β ) ( 287 528 β + 243 β 2 ) ( 605 540 β 543 β 2 + 486 β 3 ) 2 ( 1177 1020 β 1113 β 2 + 972 β 3 ) ( 12 631 + 123 238 β 363 β 2 19 008 β 3 + 8748 β 4 ) < 0

Thus, we have τ S > τ T . Tariff rate is relatively lower under an emission tax policy.

A.3.2 Input Price

w T w S = α ( 1433 1203 β 1425 β 2 + 1215 β 3 ) 2 ( 1177 1020 β 1113 β 2 + 972 β 3 ) α ( 16 052 29 817 β + 1203 β 2 + 23 517 β 3 10 935 β 4 ) 2 ( 12 631 23 238 β + 363 β 2 + 19 008 β 3 8748 β 4 )

= 9 α ( 1 β ) ( 287 528 β + 243 β 2 ) ( 307 279 β 267 β 2 + 243 β 3 ) 2 ( 1177 1020 β 1113 β 2 + 972 β 3 ) ( 12 631 + 23 238 β 363 β 2 19 008 β 3 + 8748 β 4 ) < 0

This result implies that w T < w S . Input price is relatively lower under an emission tax policy.

A.3.3 Downstream Production

q A i T q A i S = 3 α ( 81 β 89 ) ( β 1 ) 2 ( 1177 1020 β 1113 β 2 + 972 β 3 ) 3 α ( 1 β ) ( 89 81 β ) ( 10 9 β ) 2 ( 12 631 23 238 β + 363 β 2 + 19 008 β 3 8748 β 4 )

= 9 α ( β 1 ) 2 ( 287 528 β + 243 β 2 ) ( 81 β 89 ) 2 ( 1177 1020 β 1113 β 2 + 972 β 3 ) ( 12 631 + 23 238 β 363 β 2 19 008 β 3 + 8748 β 4 ) > 0

This result indicates that q A i T > q A i S . The quantity of product A is relatively higher under an emission tax policy.

q B i T q B i S = α ( 307 546 β + 243 β 2 ) 2 ( 1177 1020 β 1113 β 2 + 972 β 3 ) α ( 10 9 β ) ( 307 546 β + 243 β 2 ) 2 ( 12 631 23 238 β + 363 β 2 + 19 008 β 3 8748 β 4 )

= 3 α ( β 1 ) ( 287 528 β + 243 β 2 ) ( 307 546 β + 243 β 2 ) 2 ( 1177 1020 β 1113 β 2 + 972 β 3 ) ( 12 631 + 23 238 β 363 β 2 19 008 β 3 + 8748 β 4 ) > 0

This result indicates that q B i T > q B i S . The quantity of product B is relatively higher under an emission tax policy. Thus, we have Q T > Q S, implying that total industry output is relatively higher under an emission tax policy.

A.3.4 Upstream Foreign Monopolist’s Profit

π U T π U S = 3 α 2 ( β + 1 ) ( 287 528 β + 243 β 2 ) 2 ( 1177 1020 β 1113 β 2 + 972 β 3 ) 2 3 α 2 ( 10 9 β ) ( 287 528 β + 243 β 2 ) 2 ( 11 9 β 2 ) ( 12 631 23 238 β + 363 β 2 + 19 008 β 3 8748 β 4 ) 2

= 3 α 2 ( β 1 ) ( 287 528 β + 243 β 2 ) 2 19 073 853 β 3791 451 β 2 30 139 182 β 3 + 25 349 823 β 4 + 6694 569 β 5 14 283 297 β 6 + 4251 528 β 7 7155 971 ( 1177 1020 β 1113 β 2 + 972 β 3 ) 2 ( 12 631 + 23 238 β 363 β 2 19 008 β 3 + 8748 β 4 ) 2 > 0

We thus have π U T > π U S , which implies that foreign input monopolist makes a relatively higher profit under an emission tax policy.

A.3.5 Downstream Multiproduct Firms’ Profits

π i T π i S

= α 2 ( 88 169 242 343 β + 145 452 β 2 + 142 866 β 3 193 185 β 4 + 59 049 β 5 ) 2 ( 1177 1020 β 1113 β 2 + 972 β 3 ) 2 α 2 ( 10 9 β ) 2 93 389 262 341 β + 174 162 β 2 + 124 560 β 3 188 811 β 4 + 59 049 β 5 2 ( 12 631 23 238 β + 363 β 2 + 19 008 β 3 8748 β 4 ) 2

= 3 α 2 ( β 1 ) 2413 191 070 266 β 6008 899 915 917 β 2 + 6132 067 557 768 β 3 + 1446 864 157 548 β 4 10 034 124 430 212 β 5 + 9311 501 435 688 β 6 1365 695 378 856 β 7 3733 856 734 725 β 8 + 3139 788 503 106 β 9 1063 899 709 515 β 10 + 139 471 376 040 β 11 376 407 931 703 2 ( 1177 1020 β 1113 β 2 + 972 β 3 ) 2 ( 12 631 + 23 238 β 363 β 2 19 008 β 3 + 8748 β 4 ) 2 > 0

Thus, π i T > π i S , implying that each DMP firm’s profit is relatively higher under an emission tax policy.

A.3.6 Environmental Damage

E D T E D S = 3 α 2 ( 27 β 29 ) 2 ( β 1 ) 2 ( 1177 1020 β 1113 β 2 + 972 β 3 ) 2 3 α 2 ( 1 β ) 2 ( 10 9 β ) 2 ( 29 27 β ) 2 ( 12 631 23 238 β + 363 β 2 + 19 008 β 3 8748 β 4 ) 2

= 9 α 2 ( 1 β ) 3 ( 29 27 β ) 2 ( 287 528 β + 243 β 2 ) ( 24 401 44 031 β 1587 β 2 + 38 745 β 3 17 496 β 4 ) ( 1177 1020 β 1113 β 2 + 972 β 3 ) 2 ( 12 631 23 238 β + 363 β 2 + 19 008 β 3 8748 β 4 ) 2 > 0

MED T MED S = 2 α ( 1 β ) ( 29 27 β ) 1177 1020 β 1113 β 2 + 972 β 3 2 α ( 1 β ) ( 10 9 β ) ( 29 27 β ) 12631 8748 β 4 + 19 008 β 3 + 363 β 2 23 238 β

= 6 α ( 1 β ) 2 ( 8323 6561 β 3 + 21 303 β 2 23 061 β ) ( 1177 1020 β 1113 β 2 + 972 β 3 ) 2 ( 23 238 β 363 β 2 19 008 β 3 + 8748 β 4 12 631 ) 2 > 0

These results indicate that ED T > ED S and MED T > MED S . Thus, total (or marginal) environmental damage is relatively higher under an emission tax policy.

A.3.7 Consumer Surplus

CS T CS S

= α 2 ( 221 823 β + 116 238 β 2 + 161 334 β 3 197 559 β 4 + 59 049 β 5 + 82 769 ) ( 1177 1020 β 1113 β 2 + 972 β 3 ) 2 α 2 82 769 221 823 β + 116 238 β 2 + 161 334 β 3 197 559 β 4 + 59 049 β 5 ( 10 9 β ) 2 ( 12 631 23 238 β + 363 β 2 + 19 008 β 3 8748 β 4 ) 2

= 3 α 2 ( β 1 ) ( 287 528 β + 243 β 2 ) ( 44 031 β + 1587 β 2 38 745 β 3 + 17 496 β 4 24 401 ) ( 82 769 221 823 β + 116 238 β 2 + 161 334 β 3 197 559 β 4 + 59 049 β 5 ) ( 1177 1020 β 1113 β 2 + 972 β 3 ) 2 ( 23 238 β 363 β 2 19 008 β 3 + 8748 β 4 12 631 ) 2 > 0

Thus, CS T > CS S , implying that consumer surplus is relatively higher under an emission tax policy.

A.3.8 Domestic Welfare

SW T SW S = α 2 ( 287 528 β + 243 β 2 ) ( 1177 1020 β 1113 β 2 + 972 β 3 ) α 2 ( 10 9 β ) ( 287 528 β + 243 β 2 ) ( 12 631 23 238 β + 363 β 2 + 19 008 β 3 8748 β 4 )

= 3 α 2 ( 1 β ) ( 287 528 β + 243 β 2 ) 2 ( 1177 1020 β 1113 β 2 + 972 β 3 ) 23 238 β 363 β 2 19 008 β 3 + 8748 β 4 12 631 > 0

Thus, SW T > SW S , implying that domestic welfare is relatively higher under an emission tax policy.

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Received: 2022-07-15
Accepted: 2023-06-22
Published Online: 2023-07-07

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