Taxes versus emissions trading system: evaluating environmental policies that affect multiple types of pollution

Abstract

This paper examines the interaction of different policies used to control two types of agricultural pollution. Pollution control policy is efficient when both pollution types are controlled by taxes, although a tax increase on one type of pollution can increase the quantity of another type of pollution if farm inputs are substitutes. However, if one of the pollutions is controlled by a local emissions trading scheme, and another pollution type is taxed, then the pollution type which is taxed becomes less responsive to a change in its own tax levels. This policy scenario results in inefficient levels of environmental pollution outcomes unless the cap for the local emissions trading scheme is constantly being shifted in response to the tax.

Appendices

Appendix 1: Additional results from Sect. 3.1.1

The solution to Eq. 14 presented below is for the case that the farmer ceases using input 2 before input 1 as $P_\mathrm{G}$ increases. The solution is

$$\begin{aligned} \underline{\theta }^*_{i,j}= \left\{ \begin{array}{lr} \bar{\theta }_{i,j}+\frac{1}{2P_jx_i}A^{-1}_j(P_{\theta }+P^{'}_{E} \Phi ^1_j), &{} 0 \le P_\mathrm{G}< P_1\\ \left( \theta ^{1+}_{ij},0 \right) ',&{} P_1 \le P_\mathrm{G}< P_2\\ \left( 0,0 \right) ',&{} P_2 \le P_\mathrm{G}.\\ \end{array} \right. \end{aligned}$$

(24)

where

$$\begin{aligned} {\theta ^{1+}_{i,j}}= & {} \bar{\theta }^{1}_{ij}+\frac{P^1_{\theta }+ P_\mathrm{N}\phi ^{\rm N1}_{j}+P_{\rm G}\phi ^{\rm G1}_{j}+\alpha ^{12}_j \bar{\theta }^2_{i,j} x_i P_j}{2x_i\alpha ^{11}_{j}P_j},\\ P_1= & {} \frac{\alpha _j^{12}(P_{\theta }^1+P_\mathrm{N}\phi ^{\rm N1}_{j})-2\alpha ^{11}_{j}(P_{\theta }^2+P_\mathrm{N}\phi ^{\rm N2}_{j})-\bar{\theta }^1_{ij}(P_jx_i (4\alpha ^{11}_{j}\alpha ^{22}_{j}-(\alpha ^{12}_{j})^2))}{2\alpha ^{11}_{j}\phi ^{G2}_{j}-\alpha _j^{12}\phi ^{G1}_{j}}, \\ P_2= & {} -\frac{1}{\phi ^{\rm G1}_{j}}(P_j x_i(\alpha _j^{12}\bar{\theta }^2_{ij}+2\alpha ^{11}_{j}\bar{\theta }^1_{ij})+P^1_{\theta }+P_\mathrm{N}\phi ^{N1}_{j}). \end{aligned}$$

Since optimal emissions are $E^{*}_{ij} = \Phi ^{0}_j + \Phi ^{1}_{ij} \underline{\theta }^*_{i,j}$, the relationship between optimal emissions and pollution prices is also piecewise linear and the derivative matrix $\frac{\partial E^{*}_{ij}}{\partial P_\mathrm{E}}$ is:

$$\begin{aligned} \left[ \begin{array}{cc}\frac{\partial {\rm N}_{i,j}^*}{\partial P_\mathrm{N}} &{} \frac{\partial \mathrm{GHG}_{i,j}^*}{\partial P_\mathrm{N}} \\ \frac{\partial {\rm N}_{i,j}^*}{\partial P_\mathrm{G}} &{} \frac{\partial \mathrm{GHG}_{i,j}^*}{\partial P_\mathrm{G}}\end{array}\right] = \left\{ \begin{array}{lr} \frac{1}{2P_jx_i} {\Phi ^1_j}^{'} A^{-1}_j \Phi ^1_j, &{} 0< P_\mathrm{G}< P_1\\ \left[ \begin{array}{cc}\frac{(\phi ^{N1}_{j})^2}{2x_i\alpha ^{11}_{j}P_j} &{} \frac{\phi ^{N1}_j\phi ^{{\rm G}1}_{j}}{2x_i\alpha ^{11}_{j}P_j} \\ \frac{\phi ^{{\rm N}1}_j\phi ^{{\rm G}1}_{j}}{2x_i\alpha ^{11}_{j}P_j} &{} \frac{(\phi ^{{\rm G}1}_{j})^2}{2x_i\alpha ^{11}_{j}P_j}\end{array}\right] ,&{} P_1< P_{\rm G} < P_2\\ \left( 0,0 \right) ',&{} P_2 > P_\mathrm{G}.\\ \end{array} \right. \end{aligned}$$

(25)

The proofs of Propositions 2 and 3 come from expanding and inspecting the derivative matrix $\frac{\partial E^{*}_{ij}}{\partial P_\mathrm{E}}$. The proof of Proposition 1 has two parts. When $P_\mathrm{G} < P_1$, $\frac{\partial {\rm N}_{i,j}^*}{\partial P_\mathrm{N}}$ and $\frac{\partial \mathrm{GHG}_{i,j}^*}{\partial P_\mathrm{G}}$ are negative because the matrices A and hence $A^{-1}$ are negative definite. When $P_1 \le P_\mathrm{G} \le P_2$ and only one input $(\theta ^1_{ij})$ is used, inspection of Eq. 25 shows the derivatives are negative as $\alpha ^{11}_{j} <0$.

Appendix 2: Proof that $\frac{{\rm dGHG}^*}{{\rm d}P_{\rm G}} <0$ when there is an NTS

Proof.

$$\begin{aligned} \frac{\mathrm{dGHG}^*}{\mathrm{d}P_\mathrm{G}}= & {} \sum _{j\in \mathbf {J}}\sum _{i\in \mathbf {I}^j}\frac{\partial \mathrm{GHG}_{i,j}^*}{\partial P_\mathrm{G}}+\sum _{j\in \mathbf {J}}\sum _{i\in \mathbf {I}^j}\frac{\partial \mathrm{GHG}_{i,j}^*}{\partial P_\mathrm{N}}\times \frac{\partial P_\mathrm{N}}{\partial P_\mathrm{G}} \nonumber \\= & {} \sum _{j\in \mathbf {J}}\sum _{i\in \mathbf {I}^j}\frac{\partial \mathrm{GHG}_{i,j}^*}{\partial P_\mathrm{G}}-\sum _{j\in \mathbf {J}}\sum _{i\in \mathbf {I}^j}\frac{\partial \mathrm{GHG}_{i,j}^*}{\partial P_\mathrm{N}}\times \frac{\sum _{j\in \mathbf {J}}\sum _{i\in \mathbf {I}^j}\frac{\partial N_{i,j}^*}{\partial P_\mathrm{G}}}{\sum _{j\in \mathbf {J}}\sum _{i\in \mathbf {I}^j}\frac{\partial N_{i,j}^*}{\partial P_\mathrm{N}}} \nonumber \\= & {} \frac{\left( \sum _{j\in \mathbf {J}}\sum _{i\in \mathbf {I}^j}\frac{\partial \mathrm{GHG}_{i,j}^*}{\partial P_\mathrm{G}}\right) \left( \sum _{j\in \mathbf {J}}\sum _{i\in \mathbf {I}^j}\frac{\partial {\rm N}_{i,j}^*}{\partial P_\mathrm{N}}\right) -\left( \sum _{j\in \mathbf {J}}\sum _{i\in \mathbf {I}^j}\frac{\partial {\rm N}_{i,j}^*}{\partial P_\mathrm{G}}\right) ^2}{\sum _{j\in \mathbf {J}}\sum _{i\in \mathbf {I}^j}\frac{\partial {\rm N}_{i,j}^*}{\partial P_\mathrm{N}}} \end{aligned}$$

(26)

since $\frac{\partial P_\mathrm{N}}{\partial P_\mathrm{G}}=-\frac{\sum _{j\in \mathbf {J}}\sum _{i\in \mathbf {I}^j}\frac{\partial {\rm N}^*_{i,j}}{\partial P_\mathrm{G}}}{\sum _{j\in \mathbf {J}}\sum _{i\in \mathbf {I}^j}\frac{\partial {\rm N}^*_{i,j}}{\partial P_\mathrm{N}}}$ and $\frac{\partial {\rm N}_{i,j}^*}{\partial P_\mathrm{G}}=\frac{\partial {\rm GHG}_{i,j}^*}{\partial P_\mathrm{N}}$$\forall i$.

As the denominator in Eq. (26) is negative, $\frac{\mathrm{dGHG}^*}{\mathrm{d}P_\mathrm{G}}<0$ if the numerator is positive. The proof is done by induction by considering how the numerator changes when moving from M farmers to $M+1$ farmers. First, we show that for any individual farmer s,  $\frac{\partial \mathrm{GHG}_{s,j}^*}{\partial P_\mathrm{G}}\frac{\partial {\rm N}_{s,j}^*}{\partial P_\mathrm{N}}\ge \left( \frac{\partial {\rm N}_{s,j}^*}{\partial P_\mathrm{G}}\right) ^2$. Then, we prove

• 1. if $\left( \sum _{i=1}^M\frac{\partial \mathrm{GHG}_{i,j}^*}{\partial P_\mathrm{G}}\right) \left( \sum _{i=1}^M\frac{\partial {\rm N}_{i,j}^*}{\partial P_\mathrm{N}}\right) -\left( \sum _{i=1}^M\frac{\partial {\rm N}_{i,j}^*}{\partial P_\mathrm{G}}\right) ^2\ge 0$; and

• 2. if $\frac{\partial \mathrm{GHG}_{s,j}^*}{\partial P_\mathrm{G}}\frac{\partial {\rm N}_{s,j}^*}{\partial P_\mathrm{N}}\ge \left( \frac{\partial {\rm N}_{s,j}^*}{\partial P_\mathrm{G}}\right) ^2$ for the additional $M+1$ farmer s; then

• 3. $\left( \sum _{i=1}^{M+1}\frac{\partial \mathrm{GHG}_{i,j}^*}{\partial P_\mathrm{G}}\right) \left( \sum _{i=1}^{M+1}\frac{\partial {\rm N}_{i,j}^*}{\partial P_\mathrm{N}}\right) -\left( \sum _{i=1}^{M+1}\frac{\partial {\rm N}_{i,j}^*}{\partial P_\mathrm{G}}\right) ^2\ge 0$.

Stage 1

We show that for any farmer s, $\frac{\partial \mathrm{GHG}_{s,j}^*}{\partial P_\mathrm{G}}\frac{\partial {\rm N}_{s,j}^*}{\partial P_\mathrm{N}}\ge \left( \frac{\partial {\rm N}_{s,j}^*}{\partial P_\mathrm{G}}\right) ^2$ whether the farmer uses 1 or 2 inputs. Consider a farmer that uses only 1 input (input 2). From (18), $\frac{\partial {\rm N}^*_{s,j}}{\partial P_\mathrm{N}}=\frac{(\phi ^{\rm N2}_j)^2}{2\alpha ^{22}_jx_sP_j}$; $\frac{\partial \mathrm{GHG}^*_{s,j}}{\partial P_\mathrm{G}}=\frac{(\phi ^{\rm G2}_j)^2}{2\alpha ^{22}_jx_sP_j}$; and $\frac{\partial {\rm N}^*_{s,j}}{\partial P_\mathrm{G}}=\frac{\phi ^{\rm N2}_j\phi ^{\rm G2}_j}{2\alpha ^{22}_jx_sP_j}$. Hence,

$$\begin{aligned} \frac{\partial \mathrm{GHG}_{s,j}^*}{\partial P_\mathrm{G}}\frac{\partial N_{s,j}^*}{\partial P_\mathrm{N}}-\left( \frac{\partial N_{s,j}^*}{\partial P_\mathrm{G}}\right) ^2= & {} \frac{1}{2\alpha ^{22}_jx_sP_j}\left[ (\phi ^{G2}_j)^2(\phi ^{N2}_j)^2-(\phi ^{N2}_j\phi ^{G2}_j)^2\right] =0. \end{aligned}$$

Second, consider a farmer who uses both inputs. Using the expressions for $\frac{\partial \mathrm{GHG}_{s,j}^*}{\partial P_\mathrm{G}}$, $\frac{\partial {\rm N}_{s,j}^*}{\partial P_\mathrm{N}}$, and $\frac{\partial {\rm N}_{s,j}^*}{\partial P_\mathrm{G}}$ from Eq. 17, it can be shown that

$$\begin{aligned} \frac{\partial \mathrm{GHG}_{s,j}^*}{\partial P_\mathrm{G}}\frac{\partial {\rm N}_{s,j}^*}{\partial P_\mathrm{N}}-\left( \frac{\partial {\rm N}_{s,j}}{\partial P_\mathrm{G}}\right) ^2=\frac{[4(\alpha ^{11}_j\alpha _j^{22}-(\alpha ^{12}_j)^2)((\phi ^{{\rm N}1}_j\phi ^{{\rm G}2}_j)-(\phi ^{{\rm N}2}_j\phi ^{{\rm G}1}_j))^2 ]}{4(\alpha ^{11}_j\alpha ^{22}_j-(\alpha ^{12}_j)^2)x_sP_j} \ge 0. \end{aligned}$$

Hence, whether farmer s uses one or two inputs $\frac{\partial \mathrm{GHG}_{s,j}^*}{\partial P_\mathrm{G}}\frac{\partial {\rm N}_{s,j}^*}{\partial P_\mathrm{N}}\ge \left( \frac{\partial {\rm N}_{s,j}^*}{\partial P_\mathrm{G}}\right) ^2$.

Stage 2

To prove the second stage, suppose for some collection of M farmers,

$\left( \sum _{i=1}^M\frac{\partial \mathrm{GHG}_{i,j}^*}{\partial P_\mathrm{G}}\right) \left( \sum _{i=1}^M\frac{\partial {\rm N}_{i,j^*}}{\partial P_\mathrm{N}}\right) -\left( \sum _{i=1}^M\frac{\partial {\rm N}_{i,j^*}}{\partial P_\mathrm{G}}\right) ^2\ge 0$. Let

• 1. $A_M=\sum _{i=1}^M\frac{\partial \mathrm{GHG}_{i,j}^*}{\partial P_\mathrm{G}}$; $B_M=\sum _{i=1}^M\frac{\partial {\rm N}^*_{i,j}}{\partial P_{\rm N}}$; and $C_M=\sum _{i=1}^M\left( \frac{\partial {\rm N}^*_{i,j}}{\partial P_\mathrm{G}}\right) ^2$

• 2. $a_m=\frac{\partial \mathrm{GHG}_{s,j}^*}{\partial P_\mathrm{G}}$; $b_m=\frac{\partial {\rm N}^*_{s,j}}{\partial P_\mathrm{N}}$; and $c_m=\left( \frac{\partial {\rm N}^*_{s,j}}{\partial P_\mathrm{G}}\right) ^2$.

Note $a_mA_M\ge 0$, $b_mB_M\ge 0$, and $c_mC_M\ge 0$, and let $\epsilon =A_MB_M-C_M^2 \ge 0$, $\mu = a_mb_m-c_m^2 \ge 0$. Then,

$$\begin{aligned} (A_M+a_m)(B_M+b_m)=\,\, & {} A_MB_M+a_mb_m+a_mB_M+A_Mb_m \nonumber \\=\,\, & {} (C_M^2+\epsilon )+(c^2_m+\mu )+\frac{(c_m^2+\mu )B_M}{b_m}+\frac{(C_M^2+\epsilon )b_m}{B_M} \nonumber \\= \,\,& {} (C_M^2+\epsilon )+(c^2_m+\mu )+c_mC_M\left( \frac{c_mB_M}{C_Mb_m}+\frac{C_Mb_m}{c_mB_M}\right) +\frac{\mu B_M}{b_m}+\frac{\epsilon b_m}{B_M} \nonumber \\= \,\,& {} C_M^2+c_m^2+c_mC_M \left( \frac{c_mB_M}{C_Mb_m}+\frac{C_Mb_m}{c_mB_M}\right) +(B_M+b_m)\left( \frac{\mu }{b_m}+\frac{\epsilon }{B_M}\right) \nonumber \\\ge\,\, & {} C_M^2+c_m^2+2c_mC_M +(B_M+b_m)\left( \frac{\mu }{b_m}+\frac{\epsilon }{B_M}\right) \nonumber \\\ge \,\,& {} (C_M+c_M)^2. \end{aligned}$$

(27)

as $\left( \frac{c_mB_M}{C_Mb_m}+\frac{C_Mb_m}{c_mB_M}\right)$$\ge 2$ and $\epsilon$, $\mu \ge 0$.

Hence, if a collection of M farmers satisfies condition 1, any set of $M+1$ farmers also satisfies it. Since we showed condition 1 holds when $M=1$, by induction it must hold for any arbitrary set of farmers. Thus, the numerator of equation 19 is non-negative and so $\frac{\mathrm{d GHG}^*}{\mathrm{d} P_\mathrm{G}} \le 0$.