Thinking Local but Acting Global? The Interplay Between Local and Global Internalization of Externalities

Abstract

The paper analyzes the implications of local and global pollution when two types of abatement activities can be undertaken. One type reduces solely local pollution (e.g., use of particulate matter filters) while the other mitigates global pollution as well (e.g., application of fuel saving technologies). In the framework of a 2-country endogenous growth model, the implications of different assumptions about the degree to which global externalities are internalised are analysed. Subsequently, we derive policy rules adapted to the different scenarios.

Appendix

1.1 Scenario 2: Derivation of K _A ⁱ and g _C ⁱ Along the BGP

From the FOCs for A _{L G} ⁱ and K ⁱ, (20) and (21), we get

$$\displaystyle{ g_{\lambda ^{i}} = -(1-\alpha \gamma )Y _{K}^{i} + K_{ A}{^{i}}^{-1} }$$

(47)

while combining (20) and (22) gives

$$\displaystyle{ g_{\mu ^{i}} =\delta \frac{1} {S}P_{G}^{i}Y _{ K}^{i}\left (\gamma (1-\alpha )Y _{ K}^{i} - K_{ A}{^{i}}^{-1}\right )^{-1} + a. }$$

(48)

From differentiating (20) with respect to time, we get a second expression for the dynamics of $g_{\mu ^{i}}$

$$\displaystyle{ g_{\mu ^{i}} = g_{\lambda ^{i}} + g_{A_{LG}^{i}} - g_{K}^{i} + \frac{\gamma (1-\alpha )g_{Y }^{i} \frac{Y ^{i}} {A_{LG}^{i}} - g_{A_{LG}^{i}}} {\gamma (1-\alpha ) \frac{Y ^{i}} {A_{LG}^{i}} - 1}. }$$

(49)

Along the BGP $g_{C^{i}} = g_{Y ^{i}} = g_{K^{i}} = g_{A_{L}^{i}} = g_{A_{LG}^{i}}$ again has to hold, such that we get from (49) that along the BGP $g_{\mu ^{i}} = g_{\lambda ^{i}} + g_{K^{i}}$. Using also

$$\displaystyle{ g_{K^{i}} = (1-\alpha \gamma )Y _{K}^{i} - C_{ K}^{i} - K_{ A}^{i^{-1} } }$$

(50)

from (5) where we employed (9), we get

$$\displaystyle{ g_{\mu ^{i}} = -C_{K}^{i} }$$

(51)

and from equating (48) and (51)

$$\displaystyle{ -(C_{K}^{i} + a) \frac{S} {\delta P_{G}^{i}} = \left (\gamma (1-\alpha ) - (K_{A}^{i}Y _{ K}^{i})^{-1}\right )^{-1}. }$$

(52)

Using (26) gives

$$\displaystyle{ K_{A}^{i} = \left (\gamma (1-\alpha ) + \frac{\delta a} {C_{K}^{i} + a} \frac{K_{A}^{i}} {K_{A}^{i} + K_{A}^{j}}\right )^{-\frac{1-\alpha \gamma } {1-\gamma }}\left (\frac{p_{L}^{\gamma }S^{\delta }} {(\alpha \gamma )^{\alpha \gamma }} \right )^{ \frac{1} {1-\gamma }}. }$$

(53)

Finally, by employing (15) we can now derive (23), the capital-abatement ratio along the BGP.

To derive the optimal output-capital ratio rearrange (20) to get

$$\displaystyle{ Y _{K}^{i}K_{ A}^{i} = (\gamma (1-\alpha ))^{-1}\left (\frac{\mu ^{i}} {\lambda ^{i}}\left (\,p_{G} \frac{K_{A}^{i}} {A_{LG}^{i}}\right ) + 1\right ). }$$

(54)

Inserting this expression into (52) gives

$$\displaystyle{ \frac{\mu ^{i}} {\lambda ^{i}}\left (\,p_{G} \frac{K_{A}^{i}} {A_{LG}^{i}}\right ) = - \frac{\frac{1} {2} \frac{\delta a} {C_{K}^{i}+a}} {\frac{1} {2} \frac{\delta a} {C_{K}^{i}+a} +\gamma (1-\alpha )} }$$

(55)

such that Y _K ⁱ can be rewritten as

$$\displaystyle{ Y _{K}^{i} = \frac{1} {K_{A}^{i}}\left [ \frac{1} {\frac{1} {2} \frac{\delta a} {C_{K}^{i}+a} +\gamma (1-\alpha )}\right ]. }$$

(56)

The BGP growth rate in Scenario 2 can be derived from (8), (20) and (21) to equal

$$\displaystyle{ g_{C}^{i} = \frac{1} {\sigma } \left ((1-\alpha \gamma )Y _{K}^{i} - (K_{ A}^{i})^{-1}-\rho \right ). }$$

(57)

Using (56) we get (25) and (27) respectively.

To show that the growth rate in Scenario 2, (27), is higher than in Scenario 1, (14), consider the following: Inserting K _A ⁱ from (16) and (23) into (14) and (27) respectively shows that g _C ⁱ⁽²⁷⁾ > g _C ⁱ⁽¹⁴⁾ holds for

$$\displaystyle{ A> 1\ \text{with}\ A = \left (1 -\frac{1} {2} \frac{1} {1-\gamma } \frac{\delta a} {C_{K}^{i} + a}\right )^{1-\gamma -\delta }\left (1 + \frac{1} {2} \frac{1} {\gamma (1-\alpha )} \frac{\delta a} {C_{K}^{i} + a}\right )^{\gamma (1-\alpha )+\delta } }$$

Remembering that 0 < δ < (1 −γ), we get

$$\displaystyle\begin{array}{rcl} \lim _{\delta \rightarrow 0}A& =& 1 {}\\ \lim _{\delta \rightarrow (1-\gamma )}A& =& \left (1 + \frac{1} {2} \frac{1-\gamma } {\gamma (1-\alpha )} \frac{\delta a} {C_{K}^{i} + a}\right )^{\gamma (1-\alpha )}> 1 {}\\ \end{array}$$

with A rising monotonously in δ for δ ∈ [0, 1 −γ]:

$$\displaystyle\begin{array}{rcl} \frac{\partial A} {\partial \delta } = \frac{A \cdot \log \left [\left (\frac{1+\frac{1} {2} \frac{1} {\gamma (1-\alpha )} \frac{\delta a} {C_{K}^{i}+a}} {1-\frac{1} {2} \frac{1} {1-\gamma } \frac{\delta a} {C_{K}^{i}+a}} \right )^{\left (\gamma (1-\alpha )-\frac{1} {2} \frac{\delta a} {C_{K}^{i}+a}\right ) \left (1-\gamma +\frac{1} {2} \frac{\delta a} {C_{K}^{i}+a}\right ) }\right ]} {\left (1 -\gamma -\frac{1} {2} \frac{1} {1-\gamma } \frac{\delta a} {C_{K}^{i}+a}\right )\left (\gamma (1-\alpha ) + \frac{1} {2} \frac{\delta a} {C_{K}^{i}+a}\right )}> 0.& & {}\\ \end{array}$$

As A > 1 holds for 0 < δ < 1 −γ, internalization of the global externality raises the growth rate (i.e. g _C ⁱ⁽²⁷⁾ > g _C ⁱ⁽¹⁴⁾).

1.2 Reaction of C _K ⁱ to an Increasing Degree of Internalization of Global Damages

To derive the effect an increasing degree of internalization of the global externality has in the symmetric scenarios, consider the following slightly more general version of (30)

$$\displaystyle{ K_{A}^{i} = \left (\gamma (1-\alpha ) + \Delta \frac{\delta a} {C_{K}^{i} + a}\right )^{- \frac{1-\alpha \gamma } {1-\gamma -\delta }}\left (\frac{p_{L}^{\gamma }\left (\frac{2p_{G}} {a} \right )^{\delta }} {(\alpha \gamma )^{\alpha \gamma }} \right )^{ \frac{1} {1-\gamma -\delta }} }$$

(58)

where $0 <\Delta <1$ represents the degree of symmetric internalization of the global damages (recall that for Scenario 2, $\Delta = \frac{1} {2}$, while for Scenario 3, $\Delta = 1$, holds).

Inserting (58) into (26) gives

$$\displaystyle{ Y _{K}^{i} = \left (\gamma (1-\alpha ) + \Delta \frac{\delta a} {C_{K}^{i} + a}\right )^{\frac{(1-\alpha )\gamma +\delta } {1-\gamma -\delta } }\left (\frac{p_{L}^{\gamma }\left (\frac{2p_{G}} {a} \right )^{\delta }} {(\alpha \gamma )^{\alpha \gamma }} \right )^{- \frac{1} {1-\gamma -\delta }}. }$$

(59)

From equating

$$\displaystyle{ g_{C}^{i} = \frac{1} {\sigma } \left (\left (1 -\gamma -\Delta \frac{\delta a} {C_{K}^{i} + a}\right )Y _{K}^{i}-\rho \right ) }$$

(60)

which represents the BGP growth rate (31) for a degree of internalization of $\Delta$, to (50) under consideration of $g_{C^{i}} = g_{K^{i}}$, we get C _K ⁱ as a function of parameters only after inserting (56) and (59), where $\frac{1} {2}$ in (56) was substituted by $\frac{1} {\Delta }$:

$$\displaystyle{ C_{K}^{i} = \frac{\rho } {\sigma } + \frac{\sigma -1} {\sigma } \left (1-\gamma -\Delta \frac{\delta a} {C_{K}^{i} + a}\right )\left (\gamma (1-\alpha ) + \Delta \frac{\delta a} {C_{K}^{i} + a}\right )^{\frac{\gamma (1-\alpha )+\delta } {1-\gamma -\delta } }\Omega }$$

(61)

with $\Omega = \left (\,p_{L}^{\gamma }\left (\frac{2p_{G}} {a} \right )^{\delta }(\alpha \gamma )^{-\alpha \gamma }\right )^{- \frac{1} {1-\gamma -\delta }}$.

The reaction of the consumption-capital ratio to a marginal increase in the degree of internalization is then given by

$$\displaystyle{ \frac{\partial LHS} {\partial C_{K}^{i}} \,dC_{K}^{i} = \frac{\partial RHS} {\partial C_{K}^{i}} \,dC_{K}^{i} + \frac{\partial RHS} {\partial \Delta } \,d\Delta. }$$

(62)

As $\frac{\partial LHS} {\partial C_{K}^{i}} = 1$, we get

$$\displaystyle{ \frac{dC_{K}^{i}} {d\Delta } = \frac{\partial RHS} {\partial \Delta } \left (1 -\frac{\partial RHS} {\partial C_{K}^{i}} \right )^{-1} }$$

(63)

where

$$\displaystyle\begin{array}{rcl} \frac{\partial RHS} {\partial C_{K}^{i}} & =& -\frac{\sigma -1} {\sigma } \left [1 - \Delta \frac{a} {C_{K}^{i} + a}\right ]\Theta {}\end{array}$$

(64)

$$\displaystyle\begin{array}{rcl} \frac{\partial RHS} {\partial \Delta } & =& -\frac{C_{K}^{i} + a} {\Delta } \frac{\partial RHS} {\partial C_{K}^{i}}{}\end{array}$$

(65)

with $\Theta = \frac{(1-\alpha \gamma )\delta } {1-\gamma -\delta }\,\Omega \left (\gamma (1-\alpha ) + \Delta \frac{\delta a} {C_{K}+a}\right )^{\frac{\gamma (1-\alpha )+\delta } {1-\gamma -\delta } -1}\left (\Delta \frac{\delta a} {(C_{K}+a)^{2}} \right )> 0$. Collecting terms we get

$$\displaystyle{ \frac{dC_{K}^{i}} {d\Delta } = \frac{C_{K}^{i} + a} {\Delta } \frac{\frac{\sigma -1} {\sigma } \left [1 - \Delta \frac{a} {C_{K}^{i}+a}\right ]\Theta } {1 + \frac{\sigma -1} {\sigma } \left [1 - \Delta \frac{a} {C_{K}^{i}+a}\right ]\Theta }. }$$

(66)

The sign of (66) clearly depends on $\sigma \gtrless 1$: For σ > 1, we get an unambiguous increase in C _K ⁱ for a marginal increase in $\Delta$. For σ < 1, C _K ⁱ can increase or decrease depending on the parametrization of the model.

To see whether $\Delta \frac{\delta a} {C_{K}^{i}(\Delta )+a}$ could decrease following an increase in $\Delta$, consider that in this case

$$\displaystyle{ \frac{d \frac{\Delta \delta a} {C_{K}^{i}+a}} {d\Delta } <0\qquad \Leftrightarrow \qquad \frac{dC_{K}^{i}} {d\Delta }> \frac{(C_{K}^{i} + a)} {\Delta } }$$

(67)

would have to hold. Comparison with (66) shows that this condition would only be met for $1 + \frac{\sigma -1} {\sigma } \left [1 - \Delta \frac{a} {C_{K}^{i}+a}\right ]\Omega <0$. In this case, however, $\frac{\partial RHS} {\partial C_{K}^{i}}> 1$ in (61) such that (61) would have no interior solution for C _K ⁱ in the long-run equilibrium.

Having determined that $\Delta \frac{\delta a} {C_{K}^{i}(\Delta )+a}$ is unambiguously higher for a higher degree of internalization, it follows directly from (58) that the capital-abatement ratio is lower. From (59) and (60) we get for the reaction of the growth rate to an increase in the degree of internalization:

$$\displaystyle{ \frac{dg_{C}^{i}} {d\Delta } = \frac{\Theta } {\sigma (1-\alpha \gamma )}\left (\Delta \frac{\delta a} {(C_{K}^{i} + a)^{2}}\right )^{-1}\left [1 - \Delta \frac{a} {C_{K}^{i} + a}\right ]\frac{d\left (\Delta \frac{\delta a} {C_{K}^{i}(\Delta )+a}\right )} {d\Delta }> 0. }$$

(68)

For local pollution, $P_{L}^{i} = p_{L}\left ( \frac{K^{i}} {A_{L}^{i}}\right )^{\alpha }(K_{A}^{i})^{1-\alpha }$ we get from the FOC for A _L ⁱ, (9),

$$\displaystyle{ \frac{d \frac{K^{i}} {A_{L}^{i}}} {d\Delta } = -\frac{(Y _{K}^{i})^{-2}} {\alpha \gamma } \frac{\partial Y _{K}^{i}} {\partial K_{A}^{i}} \frac{dK_{A}^{i}} {d\left (\Delta \frac{\delta a} {C_{K}^{i}(\Delta )+a}\right )}\left [1 - \Delta \frac{a} {C_{K}^{i} + a}\right ]\frac{d\left (\Delta \frac{\delta a} {C_{K}^{i}(\Delta )+a}\right )} {d\Delta } <0. }$$

(69)

As K _A ⁱ and $\frac{K^{i}} {A_{L}^{i}}$ both decrease, local pollution falls unambiguously.

1.3 Scenario 2: Derivation of Optimal τ _G

To determine the optimal τ _G , first insert τ _L from (41) into (43) which gives

$$\displaystyle{ \tau _{G} \frac{P_{G}} {A_{LG}} = 1 - (1-\alpha )\gamma Y _{K}K_{A}. }$$

(70)

From (20) we know that $1 - (1-\alpha )\gamma Y _{K}K_{A} = -\frac{\mu }{\lambda } \frac{P_{G}} {A_{LG}}$ has to hold in the optimum. Equating the two expressions shows that the tax rate has to equal the negative ratio of the shadow prices of stock pollution and capital,

$$\displaystyle{ \tau _{G} = -\frac{\mu } {\lambda }. }$$

(71)

Equating the two expressions for g _μ from (22) and (50) gives

$$\displaystyle{ -C_{K} = \frac{\delta } {S}\left (\frac{\mu } {\lambda } \frac{1} {A_{LG}}\right )^{-1}Y _{ K}K_{A} + a }$$

(72)

which reads after some rearranging

$$\displaystyle{ -\frac{\mu } {\lambda } =\delta \frac{Y } {S} \frac{1} {C_{K} + a}. }$$

(73)

Combining (71) and (73) finally gives the optimal tax rate τ _G in (45).