Who should bear the administrative costs of an emissions tax?

Abstract

All environmental policies involve costs of implementation and management that are distinct from pollution sources’ abatement costs. In practice, regulators and sources usually share these administrative costs. We examine theoretically an optimal policy consisting of an emissions tax and the distribution of administrative costs between the government and regulated sources of pollution. Our focus is on the optimal distribution of administrative costs between polluters and the government and the optimal level of the emissions tax in relation to marginal pollution damage. We demonstrate how the policy variables affect aggregate equilibrium administrative costs and show that these effects are generally indeterminate, as is the effect of the distribution of administrative costs on aggregate emissions. Consequently, the optimal sharing of administrative costs and whether the optimal emissions tax is higher or lower than marginal damage depend on specific contexts.

Appendix

Derivations of the results in Table 2. Use (2) and (3) to obtain:

$$\begin{aligned} S\left[ \begin{array}{l} q_{t}\\ x_{t} \end{array}\right] =\left[ \begin{array}{l} 0\\ 1+(1-\alpha )m_{xt} \end{array}\right] ,\; S\left[ \begin{array}{l} q_{\alpha }\\ x_{\alpha } \end{array}\right] =\left[ \begin{array}{c} 0\\ -m_{xt} \end{array}\right] ,\; \mathrm{and}\,\quad S\left[ \begin{array}{l} q_{p}\\ x_{p} \end{array}\right] =\left[ \begin{array}{l} -1\\ 0 \end{array}\right] ,\nonumber \\ \end{aligned}$$

(27)

where

$$\begin{aligned} S=\left[ \begin{array}{lc} -c_{qq} &{} -c_{qx}\\ -c_{xq} &{} -c_{xx}-(1-\alpha )m_{xx} \end{array}\right] . \end{aligned}$$

Let $|S|=c_{qq}\left( c_{xx}+(1-\alpha )m_{xx}\right) -\left( c_{qx}\right) ^{2}$ denote the determinant of the Hessian matrix of the firm’s total cost function (production cost plus the fraction of the administrative costs it bears). $|S|>0$ is required for concavity of the firm’s profit function. In the usual manner find the following comparative statics:

$$\begin{aligned} q_{t}&\!= c_{qx}(c_{xx}\!+\!(1\!-\!\alpha )m_{xx})/|S|<0,\quad \, q_{\alpha }\!=\!-c_{qx}m_{x}/|S|\ge 0,\quad \mathrm{and}\nonumber \\&\,\,q_{p}\!=\!(c_{xx}\!+\!(1\!-\!\alpha )m_{xx})/|S|>0; \end{aligned}$$

(28)

$$\begin{aligned} x_{t}&\!= -c_{qq}(1\!+\!(1\!-\!\alpha )m_{xt})/|S|<0,\quad \, x_{\alpha }\!=\!c_{qq}m_{x}/|S|\ge 0,\quad \mathrm{and}\quad \nonumber \\&x_{p}\!=\!c_{xx}/|S|>0; \end{aligned}$$

(29)

To determine the direct effects of $t,\, \alpha$, and $p$ on the size of the industry substitute (5) into (4) to obtain $\pi (t,\alpha ,p,\lambda ^{m})\equiv 0$ and:

$$\begin{aligned} \lambda _{t}^{m}=(x+(1-\alpha )m_{t})/\pi _{\lambda }>0,\quad \lambda _{\alpha }^{m}=-m/\pi _{\lambda }\le 0,\quad \mathrm{and}\quad \lambda _{p}^{m}=-q/\pi _{\lambda }<0.\nonumber \\ \end{aligned}$$

(30)

To determine the effects of $t, \, \alpha$, and $p$ on the supply function of the regulated industry, first define:

$$\begin{aligned} q^{m}=q(t,\alpha ,p,\lambda ^{m}) \quad \mathrm{and}\quad x^{m}=x(t,\alpha ,p,\lambda ^{m}) \end{aligned}$$

(31)

as the output and emissions of the cut-off firm in the industry. Then:

$$\begin{aligned} Q_{p}^{S}(t,\alpha ,p)=\int _{\lambda ^{m}}^{\bar{\lambda }}q_{p}d\lambda -\lambda _{p}^{m}q^{m}>0; \end{aligned}$$

(32)

$$\begin{aligned} Q_{t}^{S}(t,\alpha ,p)=\int _{\lambda ^{m}}^{\bar{\lambda }}q_{t}d\lambda -\lambda _{t}^{m}q^{m}<0; \end{aligned}$$

(33)

$$\begin{aligned} Q_{\alpha }^{S}(t,\alpha ,p)=\int _{\lambda ^{m}}^{\bar{\lambda }}q_{\alpha }d\lambda -\lambda _{\alpha }^{m}q^{m}\ge 0; \end{aligned}$$

(34)

From the market-clearing condition $Q^{S}(t,\alpha ,p)=Q^{D}(p)$ obtain:

$$\begin{aligned} \hat{p}_{t}=-Q_{t}^{S}/(Q_{p}^{S}-Q_{p}^{D})>0\quad \,\mathrm{and}\quad \;\hat{p}_{\alpha }=-Q_{a}^{S}/(Q_{p}^{S}-Q_{p}^{D})\le 0. \end{aligned}$$

(35)

The sign of $\hat{p}_{t}$ follows from $Q_{p}^{S}-Q_{p}^{D}>0$ and (33). The sign of $\hat{p}_{\alpha }$ follows from $Q_{p}^{S}-Q_{p}^{D}>0$ and (34), noting the sign of $q_{\alpha }$ from (28).

The signs of $\hat{x}_{t}$ and $\hat{\lambda }_{t}^{m}$ in Table 2 are by assumption as noted in section 2.3. The sign of $\hat{x}_{\alpha }$ is demonstrated in the main text.

To show the results for $\hat{\lambda }_{\alpha }^{m}$, recall from (12) that $\hat{\lambda }_{\alpha }^{m}=\lambda _{\alpha }^{m}+\lambda _{p}^{m}\hat{p}_{\alpha }$. From Table 2, $\lambda _{\alpha }^{m}=m_{x}=0$ imply $\hat{p}_{\alpha }=0$, and $\hat{\lambda }_{\alpha }^{m}=0$. Moreover, $m_{x}>0$ and $\lambda _{\alpha }^{m}=0$ imply $\hat{p}_{\alpha }<0$. Then, since $\lambda _{p}^{m}\hat{p}_{\alpha }>0, \, m_{x}>0$ and $\lambda _{\alpha }^{m}=0$ imply $\hat{\lambda }_{\alpha }^{m}>0$. To show $\hat{\lambda }_{\alpha }^{m}<0$ when $\lambda _{\alpha }^{m}<0$ and $m_{x}=0$ use (35) and (34) to rewrite $\hat{\lambda }_{\alpha }^{m}=\lambda _{\alpha }^{m}+\lambda _{p}^{m}\hat{p}_{\alpha }$ as:

$$\begin{aligned} \hat{\lambda }_{\alpha }^{m}=\lambda _{\alpha }^{m}-\lambda _{p}^{m}\left\{ \frac{\int _{\lambda ^{m}}^{\bar{\lambda }}q_{\alpha }d\lambda -\lambda _{\alpha }^{m}q^{m}}{Q_{p}^{S}-Q_{p}^{D}}\right\} . \end{aligned}$$

Since $m_{x}=0$ implies $q_{\alpha }=0$ (see (28)):

$$\begin{aligned} \hat{\lambda }_{\alpha }^{m}=\lambda _{\alpha }^{m}\left\{ \frac{Q_{p}^{S}-Q_{p}^{D}+\lambda _{p}^{m}q^{m}}{Q_{p}^{S}-Q_{p}^{D}}\right\} . \end{aligned}$$

Substitute in for $Q_{p}^{S}$ using (32) to obtain:

$$\begin{aligned} \hat{\lambda }_{\alpha }^{m}=\lambda _{\alpha }^{m}\left\{ \frac{\int _{\lambda ^{m}}^{\bar{\lambda }}q_{p}d\lambda -Q_{p}^{D}}{Q_{p}^{S}-Q_{p}^{D}}\right\} , \end{aligned}$$

which is strictly negative when $\lambda _{\alpha }^{m}<0$, because $q_{p}>0,\, Q_{p}^{D}<0$, and $Q_{p}^{S}-Q_{p}^{D}>0$.

Derivation of first order conditions (20) and (21). From (19) the first order condition for a strictly positive tax can be written as:

$$\begin{aligned} W_{t}(t,\alpha )&= p(\hat{Q})\left( \int _{\hat{\lambda }^{m}}^{\bar{\lambda }}\hat{q_{t}}d\lambda -\hat{\lambda }_{t}^{m}\hat{q}^{m}\right) \nonumber \\&-\left( \int _{\hat{\lambda }^{m}}^{\bar{\lambda }}(c_{q}\hat{q}_{t}+c_{x}\hat{x}_{t})d\lambda -\hat{\lambda }_{t}^{m}c(\hat{q}^{m},\hat{x}^{m},\hat{\lambda }^{m})\right) \nonumber \\&-\delta \left( \int _{\hat{\lambda }^{m}}^{\bar{\lambda }}\hat{x}_{t}d\lambda -\hat{\lambda }_{t}^{m}\hat{x}^{m}\right) \nonumber \\&-(1-\alpha +\alpha \mu )\left( \int _{\hat{\lambda }^{m}}^{\bar{\lambda }}(m_{t}+m_{x}\hat{x}_{t})d\lambda -\hat{\lambda }_{t}^{m}m(t,\hat{x},\hat{\lambda }^{m})\right) =0,\qquad \end{aligned}$$

(36)

where $\hat{q}^{m}=q(t,\alpha ,\hat{p},\hat{\lambda }^{m})$ is the equilibrium output of the cut-off firm in the industry. From a firm’s first order conditions for its optimal choices of output and emissions (2), substitute $p(\hat{Q})=c_{q}$ and $t=-c_{x}-(1-\alpha )m_{x}$ into (36) and rearrange terms to obtain:

$$\begin{aligned} W_{t}(t,\alpha )&= \int _{\hat{\lambda }^{m}}^{\bar{\lambda }}\left( (t-\delta )\hat{x}_{t}-\alpha \mu m_{x}\hat{x}_{t}-(1-\alpha +\alpha \mu )m_{t}\right) d\lambda \nonumber \\&-\hat{\lambda }_{t}^{m}\left( p(\hat{Q})\hat{q}^{m}-c(\hat{q}^{m},\hat{x}^{m},\hat{\lambda }^{m})\right. \nonumber \\&\left. -\delta \hat{x}^{m}-(1-\alpha +\alpha \mu )m(t,\hat{x}^{m},\hat{\lambda }^{m})\right) =0. \end{aligned}$$

(37)

Substitute equilibrium values of the cut-off firm into (4) to obtain:

$$\begin{aligned} \pi (t,\alpha ,\hat{\lambda }^{m})=p\hat{q}^{m}-c(\hat{q}^{m},\hat{x}^{m},\hat{\lambda }^{m})-t\hat{x}^{m}-(1-\alpha )m(t,\hat{x}^{m},\hat{\lambda }^{m})\equiv 0. \end{aligned}$$

(38)

Subtract $\delta \hat{x}^{m}$ and $\alpha \mu m(t,\hat{x}^{m})$ from both sides of (38) and rearrange terms to obtain:

$$\begin{aligned}&p\hat{q}^{m}-c(\hat{q}^{m},\hat{x}^{m},\hat{\lambda }^{m})-\delta \hat{x}^{m}-(1-\alpha +\alpha \mu )m(t,\hat{x}^{m},\hat{\lambda }^{m}) \nonumber \\&=(t-\delta )\hat{x}^{m}-\alpha \mu m(t,\hat{x}^{m},\hat{\lambda }^{m}). \end{aligned}$$

(39)

Substitute (39) into (37) and rearrange terms to obtain:

$$\begin{aligned} W_{t}(t,\alpha )&= (t-\delta )\left\{ \int _{\hat{\lambda }^{m}}^{\bar{\lambda }}\hat{x}_{t}d\lambda -\hat{\lambda }_{t}^{m}\hat{x}^{m}\right\} \nonumber \\&-\alpha \mu \left\{ \int _{\hat{\lambda }^{m}}^{\bar{\lambda }}(m_{t}+m_{x}\hat{x}_{t})d\lambda -\hat{\lambda }_{t}^{m}m(t,\hat{x}^{m},\hat{\lambda }^{m})\right\} \nonumber \\&-(1-\alpha )\int _{\hat{\lambda }^{m}}^{\bar{\lambda }}m_{t}d\lambda =0. \end{aligned}$$

(40)

Substitute (15) into (40) to obtain the desired result (20).

Now turn to the first order condition for $\alpha$, which from (19) is:

$$\begin{aligned} W_{\alpha }(t,\alpha )&= p(\hat{Q})\left( \int _{\hat{\lambda }^{m}}^{\bar{\lambda }}\hat{q}_{\alpha }d\lambda -\hat{\lambda }_{\alpha }^{m}\hat{q}^{m}\right) \nonumber \\&-\left( \int _{\hat{\lambda }^{m}}^{\bar{\lambda }}(c_{q}\hat{q}_{t}+c_{x}\hat{x}_{t})d\lambda -\hat{\lambda }_{t}^{m}c(\hat{q}^{m},\hat{x}^{m},\hat{\lambda }^{m})\right) \nonumber \\&-\delta \left( \int _{\hat{\lambda }^{m}}^{\bar{\lambda }}\hat{x}_{t}d\lambda -\hat{\lambda }_{t}^{m}\hat{x}^{m}\right) \nonumber \\&-(1-\alpha +\alpha \mu )\left\{ \int _{\hat{\lambda }^{m}}^{\bar{\lambda }}m_{x}\hat{x}_{\alpha }d\lambda -\hat{\lambda }_{\alpha }^{m}m(t,\hat{x}^{m},\hat{\lambda }^{m})\right\} \nonumber \\&-(\mu -1)\int _{\hat{\lambda }^{m}}^{\bar{\lambda }}m(t,\hat{x},\lambda )d\lambda \left\{ \begin{array}{c} \le 0,\quad \,\mathrm{if}\,<0,\quad \,\mathrm{then}\quad \,\alpha =0\\ \,\ge 0,\quad \,\mathrm{if}\,>0,\quad \,\mathrm{then}\quad \,\alpha =1. \end{array}\right. \end{aligned}$$

(41)

Substitute $p(\hat{Q})=c_{q}$ and $t=-c_{x}-(1-\alpha )m_{x}$ into (41) and rearrange terms to obtain:

$$\begin{aligned} W_{\alpha }(t,\alpha )&= \int _{\hat{\lambda }^{m}}^{\bar{\lambda }}\left( (t-\delta )\hat{x}_{\alpha }-\alpha \mu m_{x}\hat{x}_{\alpha }\right) d\lambda \nonumber \\&-\hat{\lambda }_{\alpha }^{m}\left( p(\hat{Q})\hat{q}^{m}-c(\hat{q}^{m},\hat{x}^{m},\hat{\lambda }^{m})-\delta \hat{x}^{m}-(1-\alpha +\alpha \mu )m(t,\hat{x}^{m},\hat{\lambda }^{m})\right) \nonumber \\&-(\mu -1)\int _{\hat{\lambda }^{m}}^{\bar{\lambda }}m(t,\hat{x},\lambda )d\lambda \left\{ \begin{array}{c} \le 0,\quad \,\mathrm{if}\,<0,\quad \,\mathrm{then}\quad \,\alpha =0\\ \,\ge 0,\quad \,\mathrm{if}\,>0,\quad \,\mathrm{then}\quad \,\alpha =1. \end{array}\right. \end{aligned}$$

(42)

Substitute (39) into (42) and rearrange terms to obtain:

$$\begin{aligned} W_{\alpha }(t,\alpha )&= (t-\delta )\left\{ \int _{\hat{\lambda }^{m}}^{\bar{\lambda }}\hat{x}_{\alpha }d\lambda -\hat{\lambda }_{\alpha }^{m}\hat{x}^{m}\right\} \nonumber \\&-\alpha \mu \left\{ \int _{\hat{\lambda }^{m}}^{\bar{\lambda }}m_{x}\hat{x}_{\alpha }d\lambda -\hat{\lambda }_{\alpha }^{m}m(t,\hat{x}^{m},\hat{\lambda }^{m})\right\} \nonumber \\&-(\mu -1)\int _{\hat{\lambda }^{m}}^{\bar{\lambda }}m(t,\hat{x},\lambda )d\lambda \left\{ \begin{array}{c} \le 0,\quad \,\mathrm{if}\,<0,\quad \,\mathrm{then}\quad \,\alpha =0\\ \,\ge 0,\quad \,\mathrm{if}\,>0,\quad \,\mathrm{then}\quad \,\alpha =1. \end{array}\right. \end{aligned}$$

(43)

Substitute (16) into (43) to obtain (21). $\square$