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Pollution externalities and corrective taxes in a dynamic small open economy

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This study examines the effects of tax policies in a dynamic model of a polluted small open economy with two sources of flow pollution—consumption and production—controlled by consumption and income taxes. In this setting, accumulated pollution negatively impacts households’ utility. We show that in a decentralized dynamic competitive equilibrium under exogenous tax rates, whereas a permanent increase in consumption and income taxes unambiguously reduces the steady-state pollution stock, a temporary increase in these taxes may lead to more pollution in the long run. This outcome suggests that more stringent environmental policies might be ineffective if the regulation is only temporary. We also derive the socially optimal solution and examine the optimal tax paths to achieve the social optimum. If distaste and leisure effects are sufficiently strong, tax rates decrease along the optimal path as pollution increases over time, and vice versa.

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  1. For instance, according to the “2020 World Air Quality Report” released by IQAir AirVisual (, in terms of the annual PM2.5 rankings by city, 22 of the top 30 most polluted cities globally are located in India. The report also shows that Bangladesh, China, India, and Pakistan share 49 of the 50 most polluted cities worldwide.

  2. We do not consider pollution-abatement activities. Therefore, increasing consumption taxes directly decreases consumption, thereby controlling the pollution caused by consumption activities. Similarly, an increase in income taxes leads to less labor supply, and thus reduces the pollution caused by production activities.


  4. See Jensen et al. (2015) for a literature review.

  5. Bovenberg and Heijdra (1998) study the effects of environmental taxation using an overlapping generations model.

  6. Essentially, when considering a more general utility function, \(U(c_{t},l_{t},P_{t})\), we assume that \(U_{cl}=0\). Even if \(U_{cl}\ne 0\), the main results do not change provided that \(U_{ll} U_{cc} \ge U_{cl}^2\) and \(U_{lP} U_{cc}/(U_{cl} U_{cP}) \ge 1 \ge U_{lP} U_{cc}/(U_{cP} U_{ll})\) hold. These conditions guarantee the existence and saddle-point stability of a steady state.

  7. For example, Hanna and Oliva (2015) show that the closure of a large refinery in Mexico City led to a 19.7% decline in pollution (as measured by SO\(_2\)) and a 1.3 h increase in work hours per week.

  8. When \(\tau ^{i}<0\) for \(i=y,c\), the government implements a subsidy policy. In the analysis of optimal taxation (Section 5), we allow for a case where \(\tau ^c\) and \(\tau ^y\) are not necessarily positive.

  9. Throughout this paper, we assume that tax revenues are rebated to households in a lump-sum fashion, rather than being used for pollution-abatement activities. This is because using tax revenues to finance pollution abatement would be another source of distortion, which disables the dynamics of foreign assets in the decentralized market economy to mimic the dynamics of foreign assets in the centralized economy.

  10. We denote \(c^*=c(P^{*},{\bar{\lambda }},\tau ^{c})\) and \(l^*=l(P^{*},{\bar{\lambda }},\tau ^{y})\) for notational simplicity.

  11. See Online Appendix B.

  12. We can verify that the Hamiltonian is jointly concave in the control and state variables and that the maximized Hamiltonian is concave in the state variables. Thus, the Arrow–Mangasarian sufficient conditions are satisfied.

  13. The initial values of state variables (i.e., pollution stock and foreign assets) are given constants, and we assume that these values are the same in the social planner’s problem as those in the decentralized economy: \({\tilde{P}}_0=P_0\) and \({\tilde{b}}_0=b_0\).

  14. Once the steady-state levels of consumption tax and pollution stock are uniquely determined, the steady-state levels of foreign assets and its shadow value are determined by (35).

  15. From (41b), if pollution stock goes to infinity, consumption does too. In this case, the marginal utility of consumption becomes 0 because of the Inada conditions in (5c). Consequently, from (10a) and given a finite level of shadow value \({\bar{\lambda }}^{*}\), \(\tau ^{c,*} \rightarrow -1\) as \(P^{*}\rightarrow \infty\). If pollution stock goes to 0, it follows from (41b) that consumption does too. Hence, the marginal utility of consumption becomes infinity under (5c). Using (10a), we find that \(\tau ^{c,*} \rightarrow \infty\) as \(P^{*}\rightarrow 0\).

  16. Suppose that pollution stock goes to 0. Then, (47b) indicates that \(N\left( l^{*}f({\bar{k}}) \right)\) approaches 0, as does labor supply. In this case, (5c) indicates that the marginal disutility of labor supply is 0; thus, in light of (10b), \(\tau ^{y,*} \rightarrow 1\) as \(P^{*} \rightarrow 0\). By contrast, if pollution stock approaches infinity, labor supply must do so too. In this case, because the marginal disutility of labor supply is infinite in (5c), we can conclude that \(\tau ^{y,*} \rightarrow -\infty\) as \(P^{*} \rightarrow \infty\).

  17. We confirmed that our numerical analysis results are robust to using different sets of parameter values.

  18. Pollution stock can be neither 0 nor infinity. To observe this, suppose that \({\tilde{P}}^{*} \rightarrow 0\) when \({\tilde{\phi }}^{*} \rightarrow 0\). In this case, from (30a), it must hold that \({\tilde{c}}^{*}\rightarrow 0\) and \({\tilde{l}}^{*}\rightarrow 0\). This means that \(\lim _{{\tilde{c}}^{*}\rightarrow 0}u_{c}=\infty\) and \(\lim _{{\tilde{l}}^{*} \rightarrow 0}\omega _{l}=0\) in light of (5c), which contradicts (A.9) and (A.10). Similarly, suppose that \({\tilde{P}}^{*}\rightarrow \infty\). Then, from (30a), it must hold that \({\tilde{c}}^{*}\rightarrow \infty\) and/or \({\tilde{l}}^{*} \rightarrow \infty\); thus, \(\lim _{{\tilde{c}}^{*}\rightarrow \infty }u_{c}=\bar{{\tilde{\lambda }}}^{*}\) or \(\lim _{{\tilde{l}}^{*} \rightarrow \infty }\omega _{l}=\bar{{\tilde{\lambda }}}^{*}f({\bar{k}})\). However, these equations contradict (5c). Consequently, pollution stock must be finite when \({\tilde{\phi }}^{*}\rightarrow 0\).

  19. From (A.10) and (A.12), the sign of \(\omega _{l}({\tilde{l}}^{*},{\tilde{P}}^{*})\) is always non-negative.

  20. Note that \(\frac{\partial {\tilde{l}}^{*}}{\partial {\tilde{\phi }}^{*}}=0\) under \(\alpha _{y}=0\).

  21. Note that \(\frac{\partial {\tilde{c}}^{*}}{\partial {\tilde{\phi }}^{*}}=0\) under \(\alpha _{c}=0\).

  22. To simplify the notation, we omit the variables in each function.


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We thank Jun-ichi Itaya, Yunfang Hu, Rintaro Yamaguchi, and Masako Ikefuji for helpful comments on earlier versions of this paper.

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Correspondence to Akihiko Yanase.

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This work was financially supported by the Japan Society for the Promotion of Science (JSPS) Grants-in-Aid for Scientific Research (B) #20H01492. We gratefully acknowledge the support.



1.1 A.1 Proof of Proposition 3

During the period between time 0 and T, the government sets a tax rate \(\tau _1^i\), which differs from the original rate, \(\tau _0^i\), for \(i=c,y\). From time T onward, the government reverts to the original level, \(\tau _{0}^{i}\). In the following, we call the former (i.e., \(0\le t<T\)) Period 1 and the latter (i.e., \(t\ge T\)) Period 2.

During Period 1, in which the tax rates are \(\tau _1^c\) and \(\tau _1^y\), the economy moves along an unstable transitional path:

$$\begin{aligned}&P_{t}=P_{1}^{*}+R_{1}e^{\mu _{1}^{s}t}+R_{2}e^{\mu _{1}^{u}t}, \end{aligned}$$
$$\begin{aligned}&\quad b_{t}=b_{1}^{*}+\left( f({\bar{k}})\frac{\partial l_{1}^{*}}{\partial P_{1}^{*}}-\frac{\partial c_{1}^{*}}{ \partial P_{1}^{*}} \right) \left( \frac{R_{1}}{\mu _{1}^{s}-r}e^{\mu _{1}^{s}t}+\frac{R_{2}}{\mu _{1}^{u}-r}e^{\mu _{1}^{u}t} \right) , \end{aligned}$$

where \(\mu _{1}^{s}\) and \(\mu _{1}^{u}\) represent stable and unstable roots, respectively, under \((\tau _{1}^{c},\tau _{1}^{y},P_{0},b_{0})\), \(P_{1}^{*}\), \(b_{1}^{*}\) and \({\bar{\lambda }}^{*}\) are steady-state equilibrium values when the pair of taxes is given by \((\tau _1^c, \tau _1^y)\), and \(R_{1}\) and \(R_{2}\) are constants to be determined. Specifically, given the initial levels of pollution stock and foreign assets, \((b_{0},P_{0})\), \(R_1\) and \(R_2\) are determined by setting \(t=0\) in (A.1a) and (A.1b) and calculating them:

$$\begin{aligned}&R_{1}=\frac{(\mu _{1}^{u}-r)(\mu _{1}^{s}-r)}{\mu _{1}^{s}-\mu _{1}^{u}}\left( \frac{P_{0}-P_{1}^{*}}{\mu _{1}^{u}-r}-\frac{b_{0}-b_{1}^{*}}{f({\bar{k}})\frac{\partial l_{1}^{*}}{\partial P_{1}^{*}}-\frac{\partial c_{1}^{*}}{\partial P_{1}^{*}}} \right) , \end{aligned}$$
$$\begin{aligned}&\quad R_{2}=\frac{(\mu _{1}^{u}-r)(\mu _{1}^{s}-r)}{\mu _{1}^{u}-\mu _{1}^{s}}\left( \frac{P_{0}-P_{1}^{*}}{\mu _{1}^{s}-r}-\frac{b_{0}-b_{1}^{*}}{f({\bar{k}})\frac{\partial l_{1}^{*}}{\partial P_{1}^{*}}-\frac{\partial c_{1}^{*}}{\partial P_{1}^{*}}} \right) . \end{aligned}$$

Given \((b_{0},P_{0})\), the steady-state levels of pollution stock and foreign assets, \(P_{1}^{*}\) and \(b_{1}^{*}\), are determined under the new tax rates, \(\tau _{1}^{i}\) \((i=c\) or y):

$$\begin{aligned}&P_{1}^{*}=P({\bar{\lambda }}_{1},\tau _{1}^{c},\tau _{1}^{y})=P(L(\tau _{1}^{c},\tau _{1}^{y},b_{0},p_{0}),\tau _{1}^{c},\tau _{1}^{y}), \end{aligned}$$
$$\begin{aligned}&\quad b_{1}^{*}=B({\bar{\lambda }}_{1},\tau _{1}^{c},\tau _{1}^{y})=B(L(\tau _{1}^{c},\tau _{1}^{y},b_{0},p_{0}),\tau _{1}^{c},\tau _{1}^{y}), \end{aligned}$$
$$\begin{aligned}&\quad {\bar{\lambda }}_{1}=L(\tau _{1}^{c},\tau _{1}^{y},b_{0},p_{0}). \end{aligned}$$

Over this period, it must hold that \({\bar{\lambda }}_{0} \ne {\bar{\lambda }}_{1}\), because the shadow value jumps after the initial change in the tax rates.

During Period 2, in which the tax rates return to \(\tau _0^c\) and \(\tau _0^y\), the economy follows a stable path as follows:

$$\begin{aligned}&P_{t}=P_{2}^{*}+R_{1}^{\prime }e^{\mu _{2}^{s}t}, \end{aligned}$$
$$\begin{aligned}&\quad b_{t}=b_{2}^{*}+\left( f({\bar{k}})\frac{\partial l_{2}^{*}}{\partial P_{2}^{*}}-\frac{\partial c_{2}^{*}}{\partial P_{2}^{*}} \right) \frac{R_{1}^{\prime }e^{\mu _{2}^{s}t}}{\mu _{2}^{s}-r}. \end{aligned}$$

We first look at the determination of the remaining arbitrary constant \(R_{1}^{\prime }\). By using (A.1a) and (A.4a) at time T, we can show that

$$\begin{aligned} P_{2}^{*}+R_{1}^{\prime }e^{\mu _{2}^{s}T}=P_{1}^{*}+R_{1}e^{\mu _{1}^{s}T} +R_{2}e^{\mu _{1}^{u}T}, \end{aligned}$$

where \(R_{1}\) and \(R_{2}\) are given by (A.2a) and (A.2b). Therefore, \(R_{1}^{\prime }\) is derived as

$$\begin{aligned} R_{1}^{\prime }=e^{-\mu _{2}^{s}T}\left( P_{1}^{*}-P_{2}^{*}+R_{1}e^{\mu _{1}^{s}T} +R_{2}e^{\mu _{1}^{u}T} \right) . \end{aligned}$$

The steady-state levels of \(P_{2}^{*}\) and \(b_{2}^{*}\) are determined by

$$\begin{aligned}&P_{2}^{*}=P({\bar{\lambda }}_{2}^{*},\tau _{0}^{c},\tau _{0}^{y})= P(L(\tau _{0}^{c},\tau _{0}^{y},b_{T},p_{T}),\tau _{0}^{c},\tau _{0}^{y}), \end{aligned}$$
$$\begin{aligned}&\quad b_{2}^{*}=B({\bar{\lambda }}_{2}^{*},\tau _{0}^{c},\tau _{0}^{y})=B(L(\tau _{0}^{c},\tau _{0}^{y},b_{T},p_{T}),\tau _{0}^{c},\tau _{0}^{y}), \end{aligned}$$
$$\begin{aligned}&\quad {\bar{\lambda }}_{2}^{*}=L(\tau _{0}^{c},\tau _{0}^{y},b_{T},p_{T}). \end{aligned}$$

Note that the shadow value does not change, \({\bar{\lambda }}_{1}={\bar{\lambda }}_{2}\), because under the assumption of perfect foresight, the household anticipates that the tax rates will return to their original levels.

We denote the policy changes by \(d\tau ^{i}\equiv \tau _{1}^{i}-\tau _{0}^{i}\), \(i=c, y\). As \({\bar{\lambda }}_{1}={\bar{\lambda }}_{2}\), the changes in the steady-state pollution stock can be approximated as follows:

$$\begin{aligned} P_{2}^{*}-P_{1}^{*}&=P({\bar{\lambda }}_{2}^{*},\tau _{0}^{c},\tau _{0}^{y})-P({\bar{\lambda }}_{1}^{*},\tau _{1}^{c},\tau _{1}^{y}) =-P_{\tau ^{c}}d\tau ^{c}-P_{\tau ^{y}}d\tau ^{y}, \end{aligned}$$
$$\begin{aligned} P_{1}^{*}-P_{0}^{*}&=P({\bar{\lambda }}_{1}^{*},\tau _{1}^{c},\tau _{1}^{y})-P({\bar{\lambda }}_{0}^{*},\tau _{0}^{c},\tau _{0}^{y}) \nonumber \\&=P_{{\bar{\lambda }}}\left( L_{\tau ^{c}}d\tau ^{c}+L_{\tau ^{y}}d\tau ^{y} \right) +P_{\tau ^{c}}d\tau ^{c}+P_{\tau ^{y}}d\tau ^{y}. \end{aligned}$$

From (A.6a) and (A.6b), the effects of the temporary change in each tax rate on pollution can be derived as follows:

$$\begin{aligned} P_{2}^{*}-P_{0}^{*}=P_{{\bar{\lambda }}}\left( L_{\tau ^{c}}d\tau ^{c}+L_{\tau ^{y}}d\tau ^{y} \right) , \end{aligned}$$

where the sign of \(P_{{\bar{\lambda }}}\) is the same as that of \(\varLambda\) defined by (25), and \(L_{\tau ^{c}}\) and \(L_{\tau ^{y}}\) have negative and positive signs, respectively (see Online Appendix A). Thus, we obtain the statement in Proposition 3.

1.2 A.2 Proof of Proposition 4

We begin with the conditions under which the steady state is uniquely determined. In the following, we denote \({\tilde{c}}^{*}={\tilde{c}}({\tilde{P}}^{*},\bar{{\tilde{\lambda }}}^{*},{\tilde{\phi }}^{*})\) and \({\tilde{l}}^{*}={\tilde{l}}({\tilde{P}}^{*},\bar{{\tilde{\lambda }}}^{*},{\tilde{\phi }}^{*})\).

From (30a), given the shadow value \(\bar{{\tilde{\lambda }}}^{*}\), the steady-state pollution stock can be expressed as

$$\begin{aligned} {\tilde{P}}^{*}={\tilde{P}}({\tilde{\phi }}^{*}), \end{aligned}$$

which has the following derivative:

$$\begin{aligned} \frac{\partial {\tilde{P}}^{*}}{\partial {\tilde{\phi }}^{*}} =\frac{\alpha _{c}G^{\prime }({\tilde{c}}^{*})\frac{\partial {\tilde{c}}^{*}}{\partial {\tilde{\phi }}^{*}} +\alpha _{y}N^{\prime }({\tilde{l}}^{*}f({\bar{k}}))\frac{\partial {\tilde{l}}^{*}}{\partial {\tilde{\phi }}^{*}} }{\theta -\alpha _{c}G^{\prime }({\tilde{c}}^{*})\frac{\partial {\tilde{c}}^{*}}{\partial {\tilde{P}}^{*}} -\alpha _{y}N^{\prime }({\tilde{l}}^{*}f({\bar{k}}))f({\bar{k}})\frac{\partial {\tilde{l}}^{*}}{\partial {\tilde{P}}^{*}} }(<0). \end{aligned}$$

Consider the case in which \({\tilde{\phi }}^{*}\rightarrow 0\). Then, from (27a) and (27b), we have

$$\begin{aligned}&u_{c}({\tilde{c}}^{*},{\tilde{P}}^{*})=\bar{{\tilde{\lambda }}}^{*}, \end{aligned}$$
$$\begin{aligned}&\quad \omega _{l}({\tilde{l}}^{*},{\tilde{P}}^{*})=\bar{{\tilde{\lambda }}}^{*}f({\bar{k}}). \end{aligned}$$

Given the constant shadow value \(\bar{{\tilde{\lambda }}}^{*}\), the marginal utility of consumption and marginal disutility of labor supply are also constant. This means that the steady-state levels of consumption, labor supply, and pollution stock must be finite and satisfy equations (A.9), (A.10), and (30a).Footnote 18

We next consider the case in which \({\tilde{\phi }}^{*}\rightarrow \infty\). Then, (27a) and (27b) can be rewritten as

$$\begin{aligned}&u_{c}({\tilde{c}}^{*},{\tilde{P}}^{*})=\infty , \end{aligned}$$
$$\begin{aligned}&\quad \omega _{l}({\tilde{l}}^{*},{\tilde{P}}^{*})=0, \end{aligned}$$

where we assume that \({\bar{\lambda }} =\infty\) in this case.Footnote 19 Considering (A.11), we observe that the marginal utility of consumption is infinite, so that either consumption or pollution stock must be 0. Analogously, from (A.12), the marginal disutility of labor supply is 0, so that either labor supply or pollution stock must be 0. Then, we find that all levels of consumption, labor supply, and pollution stock must be 0, to satisfy (30a). Consequently, it holds that \({\tilde{P}}^{*}\rightarrow 0\) as \({\tilde{\phi }}^{*} \rightarrow \infty\).

We now substitute (A.8) into (30b) as follows:

$$\begin{aligned} \varGamma ({\tilde{\phi }}^{*})&\equiv (\rho +\theta ){\tilde{\phi }}^{*}-\left\{ \omega _{P}({\tilde{l}}({\tilde{P}}({\tilde{\phi }}^{*}),\bar{{\tilde{\lambda }}}^{*},{\tilde{\phi }}^{*}),{\tilde{P}}({\tilde{\phi }}^{*}) )-u_{P}({\tilde{c}}({\tilde{P}}({\tilde{\phi }}^{*}),\bar{{\tilde{\lambda }}}^{*},{\tilde{\phi }}^{*}),{\tilde{P}}({\tilde{\phi }}^{*})) \right\} \nonumber \\&=0. \end{aligned}$$

We can verify that \(\lim _{{\tilde{\phi }}^{*}\rightarrow 0}\varGamma ({\tilde{\phi }}^{*})<0\) and \(\lim _{{\tilde{\phi }}^{*}\rightarrow \infty }\varGamma ({\tilde{\phi }}^{*})=\infty\). Moreover, the derivative of \(\varGamma ({\tilde{\phi }}^{*})\) is calculated as:

$$\begin{aligned} \varGamma ^{\prime }({\tilde{\phi }}^{*})&=\rho +\theta -\left[ \left( \omega _{lP}({\tilde{l}}^{*},{\tilde{P}}^{*})\frac{\partial {\tilde{l}}^{*}}{\partial {\tilde{P}}^{*}} +\omega _{PP}({\tilde{l}}^{*},{\tilde{P}}^{*}) \right) \frac{\partial {\tilde{P}}^{*}}{\partial {\tilde{\phi }}^{*}}+\omega _{lP}({\tilde{l}}^{*},{\tilde{P}}^{*})\frac{\partial {\tilde{l}}^{*}}{\partial {\tilde{\phi }}^{*}} \right] \nonumber \\&\quad +\left[ \left( u_{cP}({\tilde{c}}^{*},{\tilde{P}}^{*}) \frac{\partial {\tilde{c}}^{*}}{\partial {\tilde{P}}}+u_{PP}({\tilde{c}},{\tilde{P}}) \right) \frac{\partial {\tilde{P}}^{*}}{\partial {\tilde{\phi }}^{*}}+u_{cP}({\tilde{c}},{\tilde{P}})\frac{\partial {\tilde{c}}}{\partial {\tilde{\phi }}} \right] . \end{aligned}$$

Case (i): \(\alpha _{c}>0\) and \(\alpha _{y}=0\) In this case, (A.14) can be rewritten asFootnote 20

$$\begin{aligned} \varGamma ^{\prime }({\tilde{\phi }}^{*}) =\rho +\theta -\left\{ \frac{\partial {\tilde{P}}^{*}}{\partial {\tilde{\phi }}^{*}}\varPhi _c +\frac{\theta u_{cP}({\tilde{c}}^{*},{\tilde{P}}^{*})\frac{\partial {\tilde{c}}^{*}}{\partial {\tilde{\phi }}^{*}}}{\theta - \alpha _{c}G^{\prime }({\tilde{c}}^{*})\frac{\partial {\tilde{c}}^{*}}{\partial {\tilde{P}}^{*}}}\right\} >0, \end{aligned}$$

where we use (32a). Hence, there is a unique steady-state equilibrium. The saddle-point stability of the steady-state equilibrium can also be obtained by inspecting the determinant in \(\varXi\) in (31) and using (32a):

$$\begin{aligned} \varXi =\alpha _{c}G^{\prime }({\tilde{c}}^{*})\left[ (\rho +\theta )\frac{\partial {\tilde{c}}^{*}}{\partial {\tilde{P}}^{*}}+\frac{\partial {\tilde{c}}^{*}}{\partial {\tilde{\phi }}^{*}} \varPhi _c \right] -\theta \left( \rho +\theta +u_{cP}\frac{\partial {\tilde{c}}^{*} }{\partial {\tilde{\phi }}^{*}} \right) <0. \end{aligned}$$

Case(ii): \(\alpha _{c}=0\) and \(\alpha _{y}>0\) In this case, (A.14) can be rewritten asFootnote 21

$$\begin{aligned} \varGamma ^{\prime }({\tilde{\phi }}^{*})=\rho +\theta -\left\{ \frac{\partial {\tilde{P}}^{*}}{\partial {\tilde{\phi }}^{*}} \varPhi _l +\frac{\theta \omega _{lP}({\tilde{l}}^{*},{\tilde{P}}^*)\frac{\partial {\tilde{l}}^{*}}{\partial {\tilde{\phi }}^{*}}}{\theta -\alpha _{y}N^{\prime }({\tilde{l}}^{*}f({\bar{k}}))f({\bar{k}})\frac{\partial {\tilde{l}}^{*}}{\partial {\tilde{P}}^{*}}} \right\} >0, \end{aligned}$$

where we use (32b). Thus, the steady-state equilibrium is uniquely determined. Moreover, the saddle-path stability of the steady-state equilibrium can be shown by using \(\varXi\) in (31) and (32b):

$$\begin{aligned} \varXi =\alpha _{y}N^{\prime }({\tilde{l}}^{*}f({\bar{k}}))f({\bar{k}}) \left[ (\rho +\theta )\frac{\partial {\tilde{l}}^{*}}{\partial {\tilde{P}}^{*}} +\frac{\partial {\tilde{l}}^{*}}{\partial {\tilde{\phi }}^{*}} \varPhi _l \right] -\theta \left( \rho +\theta -\omega _{lP}\frac{\partial {\tilde{l}}^{*}}{\partial {\tilde{\phi }}^{*}} \right) <0. \end{aligned}$$

1.3 A.3 Proof of Proposition 5

Given the shadow value \({\bar{\lambda }}\), we totally differentiate (41b) and obtain:Footnote 22

$$\begin{aligned} \tau ^{c,*}=\tau ^{c}(P^{*}), \end{aligned}$$


$$\begin{aligned} \frac{\partial \tau ^{c,*}}{\partial P^{*}}=\frac{\theta - \alpha _{c}G^{\prime }\frac{\partial c^{*}}{\partial P^{*}}}{ \alpha _{c}G^{\prime }\frac{\partial c^{*}}{\partial \tau ^{c,*}}}<0. \end{aligned}$$

Consider the case in which the pollution stock is 0. Then, from (41b), the value of the right-hand side is 0 and so is \(G(c^{*})\). As the consumption level is 0, we find from (5c) that \(\lim _{c^{*}\rightarrow 0}u_{c}\rightarrow \infty\), thereby showing that the consumption tax rate is infinite from (10a). In summary, it holds that \(\lim _{P^{*}\rightarrow 0}\tau ^{c}(P^{*})=\infty\). Similarly, when the pollution stock is infinite, we find from (41b) that the pollution flow and, in turn, the consumption level is infinite. By using (5c), \(\lim _{c^{*}\rightarrow \infty }u_{c}\rightarrow 0\) holds; hence, the value of the left-hand side of (10a) is 0, which implies that because the consumption tax rate is \(-1\), \(\lim _{P^{*}\rightarrow \infty }\tau ^{c}(P^{*})=-1\).

Substituting (A.15) into (41a), we obtain \(Z^{c}(P^{*})\equiv (\rho +\theta )\tau ^{c}(P^{*}){\bar{\lambda }}^{*} -\alpha _{c}G^{\prime }(c(P^{*},{\bar{\lambda }}^{*},\tau ^{c}(P^{*})))\left\{ \omega _{P}(l(P^{*},{\bar{\lambda }}^{*}), P^{*})-u_{P}(c(P^{*},{\bar{\lambda }}^{*},\tau ^{c}(P^{*})),P^{*}) \right\} =0\). We can verify that \(\lim _{P^{*}\rightarrow 0}Z^{c}(P^{*}) =\infty\), \(\lim _{P^{*}\rightarrow \infty }Z^{c}(P^{*})<0\), and \(dZ^{c}(P^{*})/dP^{*}<0\). Consequently, the steady state is uniquely determined.

1.4 A.4 Proof of Proposition 6

The proof is similar to that of Proposition 5. First, from (47b), it holds that

$$\begin{aligned} \tau ^{y,*}=\tau ^{y}(P^{*}), \end{aligned}$$


$$\begin{aligned} \frac{\partial \tau ^{y,*}}{\partial P^{*}}=\frac{\theta - \alpha _{y}N^{\prime }f\frac{\partial l^{*}}{\partial P^{*}}}{\alpha _{y}N^{\prime }f\frac{\partial l^{*}}{\partial \tau ^{y}}}<0. \end{aligned}$$

Also, when the pollution stock approaches 0, we observe from (47b) that the pollution flow, \(N\left( l^{*}f({\bar{k}}) \right)\), is 0; therefore, the labor input is also 0. Consequently, from (5c), we show that \(0={\bar{\lambda }}(1-\tau ^{y,*})f({\bar{k}})\) in (10b). Therefore, we find that \(\lim _{P^{*}\rightarrow 0}\tau ^{y}(P^{*})=-1\). Conversely, assuming that the pollution stock goes to infinity, we find that the pollution flow is infinite in (47b); hence, the labor input also becomes infinite. Therefore, from (10b), it holds that \(\infty ={\bar{\lambda }}(1-\tau ^{y,*})f({\bar{k}})\), meaning that the income tax rate approaches \(-\infty\); it holds that \(\lim _{P^{*} \rightarrow \infty }\tau ^{y}(P^{*})=-\infty\).

Finally, substituting (A.16) into (47a) yields \(Z^{y}(P^{*})\equiv (\rho +\theta )\tau ^{y}(P^{*}){\bar{\lambda }}^{*} -\alpha _{y}N^{\prime }(l(P^{*},{\bar{\lambda }},\tau ^{y}(P^{*}))f({\bar{k}}))\left\{ \omega _{P}(l(P^{*},{\bar{\lambda }}^{*},\tau ^{y}(P^{*})))-u_{P}(c(P^{*},{\bar{\lambda }}^{*}),P^{*}) \right\} =0\). We can verify that \(\lim _{P^{*}\rightarrow 0}Z^{y}(P^{*})>0\), \(\lim _{P^{*}\rightarrow \infty }Z^{y}(P^{*})=-\infty\), and \(dZ^{y}(P^{*})/dP^{*}<0\). Thus, the steady state is uniquely determined.

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Nakamoto, Y., Yanase, A. Pollution externalities and corrective taxes in a dynamic small open economy. Int Tax Public Finance 29, 667–703 (2022).

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