Environmental Policy in a Federation with Special Interest Politics and Inter-Governmental Grants

Abstract

The paper explores the potential effect of intergovernmental grants (IGG) on sub-national (local) environmental policy in a federal structure. In the model, a politically-inclined local government receives campaign contributions from the polluters’ lobby in return for lower pollution taxes. A benevolent federal government uses IGG as an incentive to reduce the resulting distortion in the local pollution tax. IGG are formulaic transfers that are conditional on pollution levels—lower pollution in a sub-national jurisdiction relative to others translates into a higher share of the grant and vice versa. In equilibrium, the grant effect reduces the distortion created in the pollution tax by the lobby effect, and may even lead to a higher than Pigouvian tax when the local government assigns a large enough weight on social welfare and/or when the grant is large enough. Further, IGG result in the tax levels of jurisdictions becoming interdependent in an interesting way. Environmental policies in two jurisdictions may become strategic complements or substitutes depending on their relative pollution levels. The possibility of strategic substitution implies that federal welfare may not increase even when environmental policy becomes stricter in one state.

Appendices

Appendix 1: Firm-level Production Decisions

From Sect. 2.1, Eqs. (2) and (3) implicitly define the equilibrium values of $x$ and $a$ as functions of the pollution tax, $t$. Using the implicit function rule for a simultaneous $2$ equation case ($F^1(x,a,t)=0$ and $F^2(x,a,t)=0$), for the system of equations represented by Eqs. (2) and (3), respectively, we get the following:

$\frac{\partial x}{\partial t} = \frac{J_1}{J}$, where $J$ is

$$\begin{aligned} J= \left| \begin{array}{l@{\quad }l} \frac{\partial ^2\Pi (t)}{\partial x^2}&{} \frac{\partial ^2 \Pi (t)}{\partial x \partial a}\\ \frac{\partial ^2\Pi (t)}{\partial a \partial x}&{} \frac{\partial ^2\Pi (t)}{\partial a^2} \\ \end{array} \right| = \left| \begin{array}{l@{\quad }l} -C_{xx} &{} -C_{xa}-t\theta _a\\ -C_{ax}-t\theta _a &{} -C_{aa}-tx\theta _{aa} \\ \end{array} \right| \end{aligned}$$

(22)

and $J_1$=

$$\begin{aligned} J_1= \left| \begin{array}{l@{\quad }l} -\frac{\partial ^2\Pi (t)}{\partial x \partial t}&{} \frac{\partial ^2 \Pi (t)}{\partial x \partial a}\\ -\frac{\partial ^2\Pi (t)}{\partial a \partial t}&{} \frac{\partial ^2\Pi (t)}{\partial a^2} \end{array} \right| = \left| \begin{array}{l@{\quad }l} \ \theta &{} -C_{xa}-t\theta _a\\ x\theta _a &{} -C_{aa}-tx\theta _{aa} \end{array} \right| \end{aligned}$$

(23)

Here, $J$ is unambiguously $>0$ from the second-order conditions for profit maximization. If the production function is sufficiently concave in abatements costs, $J_1$ can be assumed to be $<0$, implying ${\partial x}/{\partial t} <0$.

Likewise, $\frac{\partial a}{\partial t} = \frac{J_2}{J}$, where ${J_2}$, is given by:

$$\begin{aligned} J_2= \left| \begin{array}{l@{\quad }l} \frac{\partial ^2\Pi (t)}{\partial x^2}&{} -\frac{\partial ^2\Pi (t)}{\partial x \partial t}\\ \frac{\partial ^2\Pi (t)}{\partial a \partial x}&{} -\frac{\partial ^2\Pi (t)}{\partial a \partial t} \end{array} \right| = \left| \begin{array}{l@{\quad }l} \ -C_{xx} &{} \theta \\ -C_{ax}-t\theta _a &{} x\theta _a \end{array} \right| \end{aligned}$$

(24)

Once again, ${J_2}\,>0$ and $\frac{\partial a}{\partial t}>0$ if the production function is sufficiently concave in production costs.

Appendix 2.1: Comparative Statics with Respect to $T$

By totally differentiating Eq. (13), and rearranging terms, we get:

$$\begin{aligned}&\underbrace{\left( 1- \alpha n D'' \frac{\partial e}{\partial t} + \frac{2e^*T}{(e +e^*)^3}\frac{\partial e}{\partial t} - \frac{1}{\delta } + \frac{e \frac{\partial ^2 e}{\partial t^2}}{\delta (\frac{\partial e}{\partial t})^2} \right) }_\text {r} \frac{dt}{d T} \nonumber \\&\quad = \underbrace{\frac{T}{(e +e^*)^3}(e -e^*)\frac{\partial e^*}{\partial t^*}}_\text {f}\frac{dt^*}{d T} +\frac{e^*}{(e +e^*)^2} \end{aligned}$$

(25)

Similarly totally differentiating Eq. (14) and rearranging terms:

$$\begin{aligned}&\underbrace{\left( 1-\alpha ^* n^* D^{*''}\frac{\partial e^*}{\partial t^*} +\frac{2eT}{(e +e^*)^3}\frac{\partial e^*}{\partial t^*}- \frac{1}{\delta ^*}+ \frac{e^* \frac{\partial ^2 e^*}{\partial t^{*2}}}{\delta ^*(\frac{\partial e^*}{\partial t^*})^2} \right) }_\text {g} \frac{dt^*}{d T} \nonumber \\&\quad = \underbrace{\frac{T}{(e +e^*)^3}(e^* - e)\frac{\partial e}{\partial t}}_\text {h}\frac{dt}{d T} + \frac{e}{(e +e^*)^2} \end{aligned}$$

(26)

Writing the resulting system of equations in matrix notation, we get:

$$\begin{aligned} \left| \begin{array}{l@{\quad }l} r&{}-f \\ -h&{}g \end{array} \right| = \left| \begin{array}{l} \frac{dt}{dT}\\ \frac{dt^*}{dT} \end{array} \right| = \left| \begin{array}{l}\frac{e^*}{(e +e^*)^2}\\ \frac{e}{(e +e^*)^2} \end{array} \right| \end{aligned}$$

(27)

where the value of the l.h.s determinant $rg-hf$, is given by:

$$\begin{aligned} \frac{\Delta }{\delta \frac{\partial e}{\partial t}}*\frac{\Delta ^*}{\delta ^* \frac{\partial e^*}{\partial t^*}}- \frac{T}{(e +e^*)^3}(e-e^*)\frac{\partial e^*}{\partial t^*} *\frac{T}{(e +e^*)^3}(e^*- e)\frac{\partial e}{\partial t} \end{aligned}$$

(28)

This uses the fact that the term ($r$) in expression (25) can also be written as $\frac{\Delta }{\delta \frac{\partial e}{\partial t}}$ and the term ($g$) in expression (26) can be written as $\frac{\Delta ^*}{\delta ^* \frac{\partial e^*}{\partial t^*}}$, where $\Delta$ and $\Delta ^*$ are the SOC’s of the objective functions of states $1$ and $2$ respectively.

In (28), $\Delta ,\,\Delta ^*,\,\frac{\partial e}{\partial t},\,\frac{\partial e^*}{\partial t^*} <0$. Further given that $(e-e^*)$ and $(e^*-e)$ necessarily have the opposite signs, it follows that $rg-hf >0$.

Using Cramer’s Rule, we have

$$\begin{aligned} \frac{dt}{d T}&= \frac{\frac{e^*}{(e +e^*)^2}*\frac{\Delta ^*}{\delta ^* \frac{\partial e^*}{\partial t^*}} + \frac{e}{(e +e^*)^2}\frac{T}{(e +e^*)^3}(e-e^*)\frac{\partial e^*}{\partial t^*}}{rg-fh}\nonumber \\ \frac{dt^*}{d T}&= \frac{\frac{e}{(e +e^*)^2}*\frac{\Delta }{\delta \frac{\partial e}{\partial t}} + \frac{e^*}{(e +e^*)^2} *\frac{T}{(e +e^*)^3}(e^*-e)\frac{\partial e}{\partial t}}{rg-fh} \end{aligned}$$

(29)

Thus we have, for the case when $e \lessgtr e^*$:

• $\frac{dt}{d T} > 0$

• $\frac{dt^*}{d T}$ is ambivalent in sign

The r.h.s in (29) has two terms. The first term represents a (positive) direct effect of a change in $T$. The second represents a second-order effect of $T$ via a change in $t$. Assuming that first-order effect dominates, we can infer that $\frac{dt^*}{d T} >0$, though the incentive effect of the higher grant is somewhat muted for $t^*$ due to the expected increase in $t$.

Appendix 2.2: Comparative Statics with Respect to $\delta$

Totally differentiating Eqs. (13) and (14), and writing in matrix notation, we have:

$$\begin{aligned} \left| \begin{array}{l@{\quad }l} r&{}-f\\ -h&{}g \end{array} \right| = \left| \begin{array}{l} \frac{dt}{d\delta }\\ \frac{dt^*}{d\delta } \end{array} \right| = \left| \begin{array}{l} -\frac{e}{\delta ^2 \frac{\partial e}{\partial t}}\\ 0 \end{array} \right| \end{aligned}$$

(30)

Using Cramers’s rule, we can get the comparative static results with respect to $\delta$ from (22) as follows:

$$\begin{aligned} \frac{dt}{d\delta }&= \frac{-\frac{e}{\delta ^2 \frac{\partial e}{\partial t}} *\frac{\Delta ^*}{\delta ^* \frac{\partial e^*}{\partial t^*}}}{rg-fh}\\ \frac{dt^*}{d\delta }&= \frac{-\frac{e}{\delta ^2 \frac{\partial e}{\partial t}} *\frac{T}{(e +e^*)^3}(e^*-e)\frac{\partial e}{\partial t}}{rg-fh} \end{aligned}$$

Given that $rg-fh >0$, it is easily seen that $\frac{dt}{d\delta }>0$. However, $\frac{dt^*}{d\delta } \gtrless 0$ according as $e^*\lessgtr e$

Appendix 3: Conditions for Concavity of the Best Response Functions

From Eq. (13), we can derive the slope of the best response function $t=f(t,t^*)$ as follows: $\frac{d^2 f(t^*)}{dt^{*^2}}= A*(K+ L + M + N)$, where

$$\begin{aligned} A&= \frac{\delta ^2 (\frac{\partial e}{\partial t})^2}{\Delta ^2} >0;\\ K&= \frac{4(T)^2}{(e+e^*)^7}\frac{\partial e)}{\partial t}\left( \frac{\partial e^*}{\partial t^*}\right) ^2 (e-e^*)(2e^*-e)\\ L&= \frac{\Delta T}{\delta (e+e^*)^3} \frac{\partial e}{\partial t} \left[ \frac{\partial ^2 e^*}{\partial t^{*2}}(e-e^*)-\frac{2}{(e+e^*)}\left( \frac{\partial e^*}{\partial t^*}\right) ^2 (2e-e^*)\right] \\ M&= \frac{\delta (T)^2}{\Delta (e+e^*)^7}\frac{\partial e}{\partial t}\left( \frac{\partial e^*}{\partial t^*}\right) ^2 \frac{\partial ^2 e}{\partial t^2}(e-e^*)^2 \left[ \alpha n D'' - \frac{2 T e^*}{(e+e^*)^3} +\frac{2 e}{\delta \left( \frac{\partial e}{\partial t}\right) ^3} \frac{\partial ^2 e}{\partial t^2} - \frac{1}{\delta \frac{\partial e}{\partial t}}\right] \\ N&= \frac{6\delta (T)^3}{\Delta (e+e^*)^{10}}\left( \frac{\partial e^*}{\partial t^*}\right) ^2\left( \frac{\partial e}{\partial t}\right) ^3 (e-e^*)^2 \end{aligned}$$

For concavity, we need $(K+L+M+N)<0$. By inspecting the above expressions closely, we can infer that for both cases, $e<e^*$ and $e>e^*$, concavity will require that $D''$ be large, $\Delta$ be large enough and $e^* \ngtr 2e$. Similarly, from $\frac{d^2t^*}{dt^2}$, we get the necessary condition for concavity as $D^{*''}$ large, $\Delta ^*$ large enough and $e \ngtr 2e^*$. Together, the necessary conditions for the concavity of the best-response functions may be written as: $e \ngtr 2e^*;\,e^* \ngtr 2e;\,D'',\,D^{*''}$ be large and $\Delta ,\,\Delta ^*$ be large enough.