Tax Competition in Vertically Differentiated Markets with Environmentally Conscious Consumers

Abstract

This paper analyses the effects of tax competition on environmental product quality, pollution and welfare in a two-country, vertically differentiated, international duopoly, in which consumers are environmentally conscious. The firm in each country chooses first the environmental quality of its product (which reflects the emissions generated in the production process) and then the price. In equilibrium one country will be more polluted than the other because firms choose different levels of environmental quality of their products. We find that a country’s optimal commodity tax is higher if the domestic firm is the more polluting supplier. Furthermore, non-cooperative commodity tax rates are inefficiently high in equilibrium. This is because, in this framework with environmentally aware consumers, commodity taxes affect the choice of firms regarding their emissions. Therefore, a domestic tax reduction not only raises the profits of the foreign firm but also lowers its emission levels, resulting in higher welfare for the other country. We also analyse the optimal cooperative and non-cooperative commodity and emission taxes with border tax adjustments. With these two policy instruments available, commodity taxes are higher.

Appendix

1.1 Proof of Equation (10)

Maximization of (5), after substituting (9) and (3), give the following first order conditions

$$\begin{aligned} \frac{\partial \Pi _{h}}{\partial q_{hi}}= & {} \frac{1}{36}\frac{\left( q_{hi}+2t_{i}a_{i}-2a_{i}+4t_{i}-4+q_{li}\right) \left( 3q_{hi}+2t_{i}a_{i}-2a_{i}+4t_{i}-4-q_{li}\right) }{1-t_{i}}=0, \\ \frac{\partial \Pi _{l}}{\partial q_{li}}= & {} -\frac{1}{36}\frac{\left( q_{hi}+2t_{i}a_{i}+2-2a_{i}-2t_{i}+q_{li}\right) \left( 3q_{li}+2t_{i}a_{i}-2a_{i}-2t_{i}+2-q_{hi}\right) }{1-t_{i}}=0. \end{aligned}$$

Solving simultaneously the first order conditions for profit maximization gives the following three equilibria.

$$\begin{aligned} q_{hi}= & {} t_{i}^{e}+\left( 1-t_{i}\right) \left( a_{i}+\frac{7}{2}\right) ,\quad q_{li}=t_{i}^{e}+\left( 1-t_{i}\right) \left( a_{i}+\frac{1}{2}\right) \\ q_{hi}= & {} t_{i}^{e}+\left( 1-t_{i}\right) \left( a_{i}+\frac{1}{2}\right) ,\quad q_{li}=t_{i}^{e}+\left( 1-t_{i}\right) \left( a_{i}-\frac{5}{2}\right) \\ q_{hi}= & {} t_{i}^{e}+\frac{1}{4}\left( 1-t_{i}\right) \left( 4a_{i}+5\right) ,\quad q_{li}=t_{i}^{e}+\frac{1}{4}\left( 1-t_{i}\right) \left( 4a_{i}-1\right) ~\;i=C,D. \end{aligned}$$

The last pair is the only one that satisfies the second order conditions.

1.2 Proof of Equations (14) and (18)

Substituting the Eqs. (4)–(7) in (8) gives

$$\begin{aligned} W_{C}= & {} \int _{a}^{\widehat{\theta _{i}}}(\theta q_{lC}-p_{lC})\,df(\theta )+\int _{\widehat{\theta _{i}}}^{a+1}(\theta q_{hC}-p_{hC})\,df(\theta ) \\&+\,\sum _{i=C,D}\left[ \left( 1-t_{i}\right) p_{hi}-t_{i}^{e}\left( E-q_{hi}\right) -cq_{hi}^{2}\right] d_{hi} \\&+\, t_{C}\sum _{j=h,l}p_{ji}d_{ji}+t_{C}^{e}\sum _{j=h,l}\left( E-q_{ji}\right) d_{ji} \\&-\, \left[ \left( E-q_{hC}\right) d_{hC}+\left( E-q_{hD}\right) d_{hD}\right] . \end{aligned}$$

An analogous condition applies to the other country.

After using (2), (3), (9) and (10), the welfare becomes

$$\begin{aligned} W_{C}= & {} \frac{t_{D}}{8}\left( 3t_{D}-11-4a_{D}\right) +\frac{t_{C}}{2} \left( 1-\frac{25}{16}t_{C}-a_{C}\left( 1+t_{C}\left( 1+a_{C}\right) \right) \right) +a_{C}\left( 1+\frac{1}{2}a_{C}\right) \\&-\,\frac{1}{2}a_{D}-E+\frac{41}{32}\\ W_{D}= & {} \frac{t_{C}}{8}\left( 3t_{C}-5-4a_{C}\right) +\frac{t_{D}}{2}\left( \frac{5}{2}-\frac{25}{16}t_{D}-a_{D}\left( 1+t_{D}\left( 1+a_{D}\right) \right) \right) +a_{D}\left( 1+\frac{1}{2}a_{D}\right) \\&+\, \frac{1}{2}a_{C}-E+\frac{7}{32} \end{aligned}$$

We maximize the above expressions with respect to $t_{C}$ and $t_{D}.$ The solution of the first order conditions $\frac{dW_{C}}{dt_{C}}=0$ and $\frac{ dW_{D}}{dt_{D}}=0$ gives the non-cooperative taxes

$$\begin{aligned} t_{CT}=\frac{8\left( 1-a_{C}\right) }{16a_{C}\left( 1+a_{C}\right) +25} ,\quad t_{DT}=\frac{4\left( 5-2a_{D}\right) }{16a_{D}\left( 1+a_{D}\right) +25}. \end{aligned}$$

(26)

By setting $a_{C}=a_{D}=a$ in the above expression we get the non-cooperative taxes in (14).

Maximizing the joint welfare of the two countries i.e., $\underset{ t_{C},t_{D}}{\max }(W_{C}+W_{D})$, gives

$$\begin{aligned} t_{iT}^{c}=-\frac{2\left( 8a_{i}+1\right) }{16a_{i}\left( 1+a_{i}\right) +13}\quad i=C,D. \end{aligned}$$

(27)

The cooperative taxes in (18) can be obtained by setting $a_{C}=a_{D}=a.$

1.3 Proof of Proposition 2

Suppose that both countries raise their commodity tax simultaneously. By the first order conditions $\frac{dW_{C}}{dt_{C}}=0$ and $\frac{dW_{D}}{dt_{D}} =0$, a country’s increase in tax has a zero first-order impact on its welfare. However, its welfare changes from a rise in the neighboring country’s tax. As one can see from (17), $\frac{dW_{C}}{ dt_{D}}=-\left[ \frac{3}{4}(1-t_{D})+\frac{1}{2}\left( a+\frac{5}{4}\right) \right]$ and $\frac{dW_{D}}{dt_{C}}=-\left[ \frac{3}{4}(1-t_{C})+\frac{1}{2} \left( a-\frac{1}{4}\right) \right] .$ Since both tax rates are lower than one [as one can see from (14)] and $a\geqslant \frac{ 1}{4}$, both $\frac{dW_{C}}{dt_{D}}$ and $\frac{dW_{D}}{dt_{C}}$ are negative. Therefore welfare in each country is strictly increased by a reduction in tax rates. Moreover, using (13) one can show that pollution in each country is decreased by a reduction in tax rates since $\frac{\partial Z_{C}}{\partial t_{C}}+\frac{\partial Z_{C}}{\partial t_{D}} =a+\frac{5}{4}>0$ and $\frac{\partial Z_{D}}{\partial t_{D}}+\frac{\partial Z_{D}}{\partial t_{C}}=a-\frac{1}{4}>0.$

The second part of the proposition follows directly from the expressions (19) and (20).

1.4 Proof of Equation (21)

Substituting the Eqs. (4)–(7) in (8) gives

$$\begin{aligned} W_{C}= & {} \int _{a}^{\widehat{\theta _{i}}}(\theta q_{lC}-p_{lC})\,df(\theta )+\int _{\widehat{\theta _{i}}}^{a+1}(\theta q_{hC}-p_{hC})\,df(\theta ) \\&+\sum _{i=C,D}\left[ \left( 1-t_{i}\right) p_{hi}-t_{i}^{e}\left( E-q_{hi}\right) -cq_{hi}^{2}\right] d_{hi} \\&+\,t_{C}\sum _{j=h,l}p_{ji}d_{ji}+t_{C}^{e}\sum _{j=h,l}\left( E-q_{ji}\right) d_{ji} \\&-\left[ \left( E-q_{hC}\right) d_{hC}+\left( E-q_{hD}\right) d_{hD}\right] . \end{aligned}$$

An analogous condition applies to the other country. After using (2), (3), (9) and (10 ), the welfare becomes

$$\begin{aligned} W_{C}= & {} \frac{t_{D}}{8}\left( 3t_{D}-11-4a_{D}\right) +\frac{t_{C}}{2} \left( 1-\frac{25}{16}t_{C}-a_{C}\left( 1+t_{C}\left( 1+a_{C}\right) \right) \right) +a_{C}\left( 1+\frac{1}{2}a_{C}\right) \nonumber \\&-\, \frac{1}{2}a_{D}-E+\frac{41}{32}+\frac{t_{C}^{e}}{2}\left( 1-t_{C}^{e}+t_{C}\left( 1+2a_{C}\right) \right) +\frac{t_{D}^{e}}{2}, \end{aligned}$$

(28)

$$\begin{aligned} W_{D}= & {} \frac{t_{C}}{8}\left( 3t_{C}-5-4a_{C}\right) +\frac{t_{D}}{2}\left( \frac{5}{2}-\frac{25}{16}t_{D}-a_{D}\left( 1+t_{D}\left( 1+a_{D}\right) \right) \right) +a_{D}\left( 1+\frac{1}{2}a_{D}\right) \nonumber \\&+\,\frac{1}{2}a_{C}-E+\frac{7}{32}+\frac{t_{D}^{e}}{2}\left( 1-t_{D}^{e}+t_{D}\left( 1+2a_{D}\right) \right) +\frac{t_{C}^{e}}{2}. \end{aligned}$$

(29)

To obtain the non-cooperative emissions tax $t_{i}^{e}$ and the non-cooperative commodity tax $t_{i},$ each country solves $\underset{t_{i},t_{i}^{e}}{\max }(W_{i})$. The first-order conditions for the choice of $t_{C}$ and $t_{C}^{e}$ are

$$\begin{aligned} t_{C}= & {} \frac{t_{C}^{e}\left( \frac{1}{2}+a_{C}\right) +\frac{1}{2} \left( 1-a_{C}\right) }{a_{C}+a_{C}^{2}+\frac{25}{16}},\\ t_{C}^{e}= & {} t_{C}\left( \frac{1}{2}+a_{C}\right) +\frac{1}{2}. \end{aligned}$$

The first-order conditions for the choice of $t_{D}$ and $t_{D}^{e}$ are

$$\begin{aligned} t_{D}= & {} \frac{t_{D}^{e}\left( \frac{1}{2}+a_{D}\right) +\frac{1}{2}\left( \frac{5}{2}-a_{D}\right) }{a_{D}+a_{D}^{2}+\frac{25}{16}},\\ t_{D}^{e}= & {} t_{D}\left( \frac{1}{2}+a_{D}\right) +\frac{1}{2}. \end{aligned}$$

Solving simultaneously the first-order conditions we get

$$\begin{aligned} t_{C}=\frac{4}{7},t_{D}=\frac{8}{7},t_{C}^{e}=\frac{8a_{C}+11}{14},t_{D}^{e}= \frac{16a_{D}+15}{14}. \end{aligned}$$

(30)

By setting $a_{C}=a_{D}=a$ we can find the optimal non-cooperative taxes in (21).

1.5 Proof of Equations (22) and (25)

In the cooperative equilibrium both countries choose their taxes to maximize the joint welfare i.e., $\underset{t_{C},t_{D},t_{C}^{e},t_{D}^{e}}{\max } (W_{C}+W_{D}).$ Adding (28) and (29) and maximizing it with respect to $t_{C},t_{D},t_{C}^{e},t_{D}^{e}$ yields

$$\begin{aligned} t_{C}= & {} \frac{t_{C}^{e}\left( \frac{1}{2}+a_{C}\right) -\frac{1}{8}-a_{C}}{ a_{C}+a_{C}^{2}+\frac{13}{16}}\\ t_{D}= & {} \frac{t_{D}^{e}\left( \frac{1}{2}+a_{D}\right) -\frac{1}{8}-a_{D}}{ a_{D}+a_{D}^{2}+\frac{13}{16}}\\ t_{C}^{e}= & {} t_{C}\left( \frac{1}{2}+a_{C}\right) +1\\ t_{D}^{e}= & {} t_{D}\left( \frac{1}{2}+a_{D}\right) +1 \end{aligned}$$

We solve the first order conditions simultaneously to obtain Eq. (25)

$$\begin{aligned} t_{i}^{c}=\frac{2}{3},\quad t_{i}^{ec}=\frac{2a_{i}+4}{3}\quad i=C,D \end{aligned}$$

and after setting $a_{C}=a_{D}=a$ we get the cooperative taxes in (22).

1.6 Proof of Proposition 4

Using (21) and (22), we can show that

$$\begin{aligned} t_{C}^{e}< & {} ~t_{C}^{ec}\,\text {since}\,t_{C}^{e}-~t_{C}^{ec}=-\frac{23}{42}- \frac{2a}{21}<0,\\ t_{C}< & {} ~t_{C}^{c}\,\text {since}\,t_{C}-t_{C}^{c}=-\frac{2}{21}<0,\\ t_{D}> & {} t_{D}^{c}\,\text {since}\,t_{D}-t_{D}^{c}=\frac{10}{21}>0,\\ t_{D}^{e}< & {} t_{D}^{ec}\text { if }a<\frac{11}{20},\,\text {since}\, t_{D}^{ec}-t_{D}^{e}=\frac{11-20a}{42}. \end{aligned}$$

1.7 Partially Covered Market: Numerical Example

When the market is not covered there is a consumer in each country, denoted by $\tilde{\theta }_{i}$, who is indifferent between buying the low-quality version of the product and not buying it at all, therefore $u_{li}=0.$ Following (1), one obtains

$$\begin{aligned} \tilde{\theta }_{i}=\frac{p_{li}}{q_{li}}~i=C,D. \end{aligned}$$

(31)

The demand functions are given by

$$\begin{aligned} d_{hi}=a_{i}+1-\widehat{\theta _{i}},\quad d_{li}=\widehat{\theta }_{i}-\tilde{ \theta }_{i},\quad i=C,D \end{aligned}$$

(32)

We denote $\tau _{i}=\frac{1}{1-t_{i}}$ to simplify calculations. In the case of commodity tax competition, the profit functions for the high-and low-quality firms are given by

$$\begin{aligned} \Pi _{j}=\sum _{i=C,D}\left( \frac{p_{ji}}{\tau _{i}}-cq_{ji}^{2}\right) d_{ji}\quad j=h,l. \end{aligned}$$

(33)

Maximizing profits with respect to prices, given quality levels and tax rates gives

$$\begin{aligned} p_{hi}= & {} \frac{q_{hi}}{2}\frac{4\left( q_{hi}-q_{li}\right) \left( a_{i}+1\right) +q_{li}^{2}\tau _{i}+2q_{hi}^{2}\tau _{i}}{\left( 4q_{hi}-q_{li}\right) } \quad i=C,D,\nonumber \\ p_{li}= & {} \frac{q_{li}}{2}\frac{2\left( q_{hi}-q_{li}\right) \left( a_{i}+1\right) +2q_{hi}q_{li}\tau _{i}+q_{hi}^{2}\tau _{i}}{\left( 4q_{hi}-q_{li}\right) } \quad i=C,D. \end{aligned}$$

(34)

Suppose now that $a_{C}=a_{D}=0$ and $E=2$. Maximizing profits with respect to qualities and solving simultaneously the first-order conditions gives the environmental product quality levels

$$\begin{aligned} q_{hi}=\frac{0.82}{\tau _{i}},\quad q_{li}=\frac{0.4}{\tau _{i}}\quad i=C,D. \end{aligned}$$

(35)

Following the same steps as in Sect. 3.1, after using the equations above, we obtain the following optimal cooperative and non-cooperative tax rates: $t_{DT}=-0.2,t_{CT}=-0.6$ and $t_{DT}^{c}=t_{CT}^{c}=-1.5$.

1.8 Cournot Competition: Numerical Example

In the case of Cournot competition and non covered market we get the inverse demand functions, after using (31) and (2) in (32)

$$\begin{aligned} p_{hi}=q_{hi}\left( a+1\right) -d_{hi}q_{hi}-d_{li}q_{li},\quad p_{li}=q_{li}\left( a+1\right) -d_{hi}q_{li}-d_{li}q_{li},\quad i=C,D. \end{aligned}$$

(36)

It can be easily shown that maximizing (33) with respect to quantities after using (36) gives

$$\begin{aligned} d_{hi}= & {} -\frac{1}{2}\frac{-\left( 4q_{hi}-2q_{li}\right) \left( a_{i}+1\right) -q_{li}^{2}\tau _{i}+2q_{hi}^{2}\tau _{i}}{\left( 4q_{hi}-q_{li}\right) } \quad i=C,D,\nonumber \\ d_{li}= & {} \frac{q_{hi}}{2}\frac{2\left( a_{i}+1\right) -2q_{li}\tau _{i}+q_{hi}\tau _{i}}{\left( 4q_{hi}-q_{li}\right) }\quad i=C,D. \end{aligned}$$

(37)

Suppose now that the parameters are $a_{C}=a_{D}=0$ and $E=2$. One can find the solution of the second stage of the game by substituting (37) in (33) and then maximizing with respect to the environmental product qualities. Solving simultaneously the first-order conditions gives

$$\begin{aligned} q_{hi}=\frac{0.74}{\tau _{i}},\quad q_{li}=\frac{0.59}{\tau _{i}}\quad i=C,D. \end{aligned}$$

(38)

Following the same steps as in Sect. 3.1, after using the equations above, we obtain the following optimal cooperative and non-cooperative tax rates: $t_{DT}=-0.15,t_{CT}=-0.09$ and $t_{DT}^{c}=t_{CT}^{c}=-1.26$.

1.9 Case 1 Firms produce the high environmental quality good for domestic consumption and the low environmental quality good for their exports

Suppose that the two firms produce the high environmental quality good for domestic consumption and the low environmental quality good for their exports. Let $\Pi _{C}$ denote the profits of the firm located in country C and $\Pi _{D}$ the profits of the firm located in country D. Then the profits are

$$\begin{aligned} \Pi _{C}= & {} \left[ \left( 1-t_{C}\right) p_{hC}-cq_{hC}^{2}\right] d_{hC}+\left[ \left( 1-t_{D}\right) p_{lD}-cq_{lD}^{2}\right] d_{lD},\nonumber \\ \Pi _{D}= & {} \left[ \left( 1-t_{D}\right) p_{hD}-cq_{hD}^{2}\right] d_{hD}+\left[ \left( 1-t_{C}\right) p_{lC}-cq_{lC}^{2}\right] d_{lC}. \end{aligned}$$

(39)

Moreover the total pollution that is generated by the production in each country is

$$\begin{aligned} Z_{C}=\left( E-q_{hC}\right) d_{hC}+\left( E-q_{lD}\right) d_{lD},\quad Z_{D}=\left( E-q_{lC}\right) d_{lC}+\left( E-q_{hD}\right) d_{hD}~ \end{aligned}$$

(40)

The solution of the last stage, in which firms maximize profits with respect to prices, given quality levels and tax rates, is the same as in Eq. (9). In the second stage we maximize (39) with respect to the quality levels. Solving simultaneously the first order conditions $\frac{\partial \Pi _{C}}{\partial q_{hC}}=0$, $\frac{ \partial \Pi _{D}}{\partial q_{lC}}=0$ and $\frac{\partial \Pi _{C}}{ \partial q_{lD}}=0$, $\frac{\partial \Pi _{D}}{\partial q_{hD}}=0$ gives the same environmental product quality levels as in (10). The optimal non-cooperative tax rates are determined by maximizing national welfare, i.e., $\underset{t_{C}}{\max }W_{C}$ and $\underset{t_{D}}{\max } W_{D},$ after using (39), (40), (6) and (7)

$$\begin{aligned} t_{CT}=t_{DT}=\frac{8\left( 1-a\right) }{16a\left( 1+a\right) +25}. \end{aligned}$$

By maximizing the joint welfare of the two countries, it is straightforward to derive the cooperative taxes which are the same as in (18).

1.10 Case 2 Firms produce the low environmental quality good for domestic consumption and the high environmental quality good for their exports

The profits in this case are

$$\begin{aligned} \Pi _{C}= & {} \left[ \left( 1-t_{C}\right) p_{lC}-cq_{lC}^{2}\right] d_{lC}+\left[ \left( 1-t_{D}\right) p_{hD}-cq_{hD}^{2}\right] d_{hD},\\ \Pi _{D}= & {} \left[ \left( 1-t_{D}\right) p_{lD}-cq_{lD}^{2}\right] d_{lD}+\left[ \left( 1-t_{C}\right) p_{hC}-cq_{hC}^{2}\right] d_{hC}. \end{aligned}$$

The total pollution that is generated by the production in each country is

$$\begin{aligned} Z_{C}=\left( E-q_{lC}\right) d_{lC}+\left( E-q_{hD}\right) d_{hD},\quad Z_{D}=\left( E-q_{hC}\right) d_{hC}+\left( E-q_{lD}\right) d_{lD} \end{aligned}$$

Following the same steps as in Case 1, one can find the optimal non-cooperative tax rates

$$\begin{aligned} t_{CT}=t_{DT}=\frac{4\left( 5-2a\right) }{16a\left( 1+a\right) +25}. \end{aligned}$$

while the cooperative taxes are the same as in (18).