Abstract
The paper considers a discrete continuous model where consumers choose quality of the product they buy as well as its usage. The product has two quality dimensions, intrinsic quality and environmental quality, that are in conflict with each other. It analyzes a twostage game in a vertically differentiated duopoly market, where firms choose intrinsic quality in the first stage, and compete in prices in the second stage. It examines the effects of green network, and environmental regulation in the form of an emission tax on equilibrium qualities, market share, and total emissions. It shows that while both green network effect and environmental regulation, individually, improve the overall environmental quality, the effect is stronger when the tax is imposed in the presence of green network effect. Though an increase in green network effect reduces environmental quality of both firms, the market share of the cleaner firm rises at the expense of the other firm, resulting in an overall improvement of the environment. In the presence of green network effect, an emission tax improves environmental quality of both firms with market shares unaltered, thereby resulting in a reduction in total emissions. The green network effect enhances the effect of an emission tax. We also find that the environmental friendly firm benefits from the green network effect. The optimal tax is increasing in the network effect.
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Notes
 1.
Refer to Matsukawa (2012) for further details.
 2.
 3.
This is the unique solution satisfying the second order conditions.
 4.
\(Y_i \equiv y  p_i\) or \(y \equiv Y_i + p_i\) and \(Y_i\) is the amount spent on fuel and composite good which is equal to \((\omega + t_e)x_i + p_z z_i\).
 5.
For \(\alpha > 3\gamma \lambda /b(1+4\lambda \gamma )\), the optimal emission tax is positive. This is consistent with the previous assumptions \(\lambda (2\gamma + \omega ) < 2\) and \(\alpha < (9/16b)\).
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Appendices
Appendix
Properties of indirect utility function
The conditional indirect utility function is nonincreasing in price of the fuel (\(\omega \)), price of the composite good (\(p_z\)) and nondecreasing in income (y) 
The conditional indirect utility function is quasiconvex in prices. We use expenditure function to calculate diagonal elements of Slutsky matrix. Rearranging terms in equation (8) and using \(Y_i \equiv y  p_i\), we get expenditure function as
The diagonal elements of Slutsky equation, \(s_{11}\) and \(s_{22}\) are given by
Thus all the properties of an indirect utility function are satisfied.
Conditions for fully covered market
For the market to be fully covered, all consumers including the consumer with the lowest preference parameter \(\theta = 0\) should derive a positive utility from buying a unit of the product in equilibrium. Plugging \(\theta = 0\) in equation (8), the utility is
Substituting the equilibrium values from equation (14),(15) in equation (24), we get the utility which consumer with \(\theta =0\) obtains from buying from firm L under the benchmark case (in the absence of regulation and green network effect) 
Assuming \(\delta \) is sufficiently small, the condition \(\lambda (2\gamma + \omega ) < 2\) ensures that market is fully covered.
Similarly, we can derive a condition for the market to be fully covered under environmental regulation in the presence of green network effect. Substituting the equilibrium values from equation (16), (17) in equation (24), we get utility function as
The equation (25) ensures that market is fully covered, i.e., for sufficiently small values of \(\delta \) and \(\lambda (\omega + t_e)\).
Second order conditions for profit maximization
Absence of green network effect and no regulation
The second order conditions at the market equilibrium are given by (using equation (13), (14) and \(\alpha = t_e = 0\))
The second order conditions holds under A2.
Green network effect with environmental regulation
Using equation (13), (16) we get second order conditions at the market equilibrium as
The above condition for high quality firm holds for \(\alpha < 9/16b\). The second order conditions is same for green network with no regulation.
Proof of proposition 1
Impact of green network effect with emission tax

(i)
From equation (16), we have \(\frac{ds_i^{**}}{d\alpha } = \frac{3}{(3  4 b \alpha )^2} > 0\)

(ii)
From equation (17),we have
$$\begin{aligned} \frac{dp_H^{**}}{d\alpha }= & {} \frac{2}{3} + \frac{3(9  8b\alpha )}{4(3  4b\alpha )^3}  \frac{3\lambda (2\gamma + \omega + t_e)}{(3  4b\alpha )^2}> 0 \\ \frac{dp_L^{**}}{d\alpha }= & {} \frac{1}{3} + \frac{9}{4(3  4b\alpha )^3}  \frac{3\lambda (2\gamma + \omega + t_e)}{(3  4b\alpha )^2}> 0 \end{aligned}$$ 
(iii)
From equation (17),we have
$$\begin{aligned} \frac{dq_H^{**}}{d\alpha }= & {} \frac{2b}{(3  4b\alpha )^2} < 0 \\ \frac{dq_L^{**}}{d\alpha }= & {} \frac{2b}{(3  4b\alpha )^2} > 0 \end{aligned}$$ 
(iv)
From equation (17),we have
$$\begin{aligned} \frac{d\pi _H^{**}}{d\alpha }= & {} \frac{(9  16b\alpha ) (63  108b\alpha + 64 (b\alpha )^2)}{36(3  4b\alpha )^3} < 0 \\ \frac{d\pi _L^{**}}{d\alpha }= & {} \frac{(9 8b\alpha )( 9 + 36b\alpha  32(b\alpha )^2)}{36(3  4b\alpha )^3} > 0 \end{aligned}$$ 
(v)
From equation (16), we have \(\Delta s^{**} = s_H^{**}  s_L^{**} = 3/2b\)

(vi)
By substituting \(t_e = 0\) in equation (16) and (17), it can be seen that above results hold for green network effect without environmental regulation.
Proof of proposition 2
Impact of emission tax in the absence of green network effect
The values \(\hat{s_i}\), \(\hat{p_i}\), \(\hat{q_i}\) and \(\hat{\pi _i}\) denote qualities, prices, quantity and profits under absence of green network without environmental regulation. It is calculated by replacing \(\omega \) with \((\omega + t_e)\) in equations (14) and (15).

(i)
From equation (14), we have \(\frac{d\hat{s_H}}{dt_e} = \frac{2\lambda }{3b} < 0\) and \(\frac{d\hat{s_L}}{dt_e} = 0.\)

(ii)
From equation (15), we have \(\frac{d\hat{p_H}}{dt_e} = \frac{20\lambda (2  \lambda (2\gamma + \omega + t_e))}{27b} < 0\) and \(\frac{d\hat{p_L}}{dt_e} = \frac{2\lambda (1 + 4\lambda (2\gamma + \omega + t_e))}{27b} < 0\).

(iii)
From equation (15), we have \(\frac{d\hat{q_H}}{dt_e} = \frac{2\lambda }{9} < 0\) and \(\frac{d\hat{q_L}}{dt_e} = \frac{2\lambda }{9} > 0\).

(iv)
From equation (15), we have \(\frac{d\hat{\pi _H}}{dt_e} = \frac{6\lambda (2  \lambda (2\gamma + \omega + t_e))^2}{2187b} < 0\) and \(\frac{d\hat{\pi _L}}{dt_e} = \frac{\lambda (1  2\lambda (2\gamma + \omega + t_e))(5 + 2\lambda (2\gamma + \omega + t_e))}{1458b} \gtrless 0\) for \(\lambda (2\gamma + \omega + t_e) \lessgtr 1/2\)

(v)
From equation (14), \(\Delta {\hat{s}} \equiv (\hat{s_H}  \hat{s_L}) = \frac{4  2\lambda (2\gamma + \omega + t_e)}{3b}\) and \(\frac{\Delta {\hat{s}}}{dt_e} = \frac{2\lambda }{3b} < 0\)
Proof of proposition 3
Impact of emission tax under green network

(i)
From equation (16),we have \(\frac{ds_H^{**}}{dt_e} = \frac{ds_L^{**}}{dt_e} = \frac{\lambda }{b} < 0\).

(ii)
Change in the prices can be written as
$$\begin{aligned} \frac{dp_i^{**}}{dt_e}= & {} \frac{\partial p_i^{**}}{\partial s_H^{**}} \frac{ds_H^{**}}{dt_e} + \frac{\partial p_i^{**}}{\partial s_L^{**}} \frac{ds_L^{**}}{dt_e} + \frac{\partial p_i^{**}}{\partial t_e}, i = H,L \end{aligned}$$Using equation (10) and \(\frac{ds_H^{**}}{dt_e} = \frac{ds_L^{**}}{dt_e} = \frac{\lambda }{b}\), we have
$$\begin{aligned} \frac{dp_H^{**}}{dt_e}= & {} \frac{b(2s_H^{**} + s_L^{**})\frac{ds_H^{**}}{dt_e}}{3}  \frac{\lambda \Delta s^{**}}{3} = \lambda s_H^{**}< 0 \\ \frac{dp_L^{**}}{dt_e}= & {} \frac{b(s_H^{**} + 2s_L^{**})\frac{ds_H^{**}}{dt_e}}{3} + \frac{\lambda \Delta s^{**}}{3} = \lambda s_L^{**} < 0 \end{aligned}$$ 
(iii)
Change in the quantity can be written as
$$\begin{aligned} \frac{dq_i^{**}}{dt_e}= & {} \frac{\partial q_i^{**}}{\partial s_H^{**}} \frac{ds_H^{**}}{dt_e} + \frac{\partial q_i^{**}}{\partial s_L^{**}} \frac{ds_L^{**}}{dt_e} + \frac{\partial q_i^{**}}{\partial t_e}, i = H,L \end{aligned}$$Using equation (11) and \(\frac{ds_H^{**}}{dt_e} = \frac{ds_L^{**}}{dt_e} = \frac{\lambda }{b}\), we have
$$\begin{aligned} \frac{dq_H^{**}}{dt_e}= & {} \frac{b\Delta s^{**}\frac{ds_H^{**}}{dt_e}}{3(\Delta s^{**}  \alpha )}  \frac{\lambda \Delta s^{**}}{3(\Delta s^{**}  \alpha )} = 0 \\ \frac{dq_L^{**}}{dt_e}= & {} \frac{b\Delta s^{**}\frac{ds_H^{**}}{dt_e}}{3(\Delta s^{**}  \alpha )} + \frac{\lambda \Delta s^{**}}{3(\Delta s^{**}  \alpha )} = 0 \end{aligned}$$ 
(iv)
Change in the profits can be written as
$$\begin{aligned} \frac{d\pi _i^{**}}{dt_e}= & {} \frac{\partial \pi _i^{**}}{\partial s_H^{**}} \frac{ds_H^{**}}{dt_e} + \frac{\partial \pi _i^{**}}{\partial s_L^{**}} \frac{ds_L^{**}}{dt_e} + \frac{\partial \pi _i^{**}}{\partial t_e}, i = H,L \end{aligned}$$Using equation (12) and \(\frac{ds_H^{**}}{dt_e} = \frac{ds_L^{**}}{dt_e} = \frac{\lambda }{b}\), we have
$$\begin{aligned} \frac{d\pi _H^{**}}{dt_e}= & {} \frac{d\pi _L^{**}}{dt_e} = 0 \end{aligned}$$ 
(v)
Using equation (16), \(\Delta {s}^{**} \equiv ({s_H}^{**}  {s_L}^{**}) = \frac{3}{2b}\)
Proof of proposition 4
The total emissions are given by \(E = e_Hq_H + e_Lq_L\). Using \(p_z = 1\) and \(e_i = x_i = \lambda s_i  \beta (\omega + t_e) + \delta \), we get \(E = \lambda (s_H q_H + s_L q_L)  \beta (\omega + t_e) + \delta \). We calculate total emissions by substituting the values of \(s_H\), \(s_L\), \(q_H\) and \(q_L\) for each case discussed below.
Benchmark case absence of green network effect and no regulation
Using equation (14) and (15) we get total emissions as
Environmental regulation in absence of green network effect
Replacing \(\omega \) with (\(\omega + t_e)\) in benchmark case, we get total emissions as
Green network effect without environmental regulation
Using equations (16), (17) and \(t_e = 0\) we get total emissions as
Green network effect with environmental regulation
Using equations (16) and (17) we get total emissions as
It can be clearly seen that \(E^{*} > \hat{E}\) and \(\widetilde{E} > E^{**}\). For sufficiently small values of \(\beta \) and \(3/8b< \alpha < 9/16b\) we observe that \(\hat{E} > \widetilde{E}\). Thus, \(E^{*}> \hat{E}> \widetilde{E} > E^{**}.\)
Proof of proposition 6
Optimal tax rate under green network effect with environmental regulation
Using equation (22) and substituting the equilibrium values of \(s_H\), \(s_L\) and \(\theta _2 = q_L\) from equation (16, 17) we get 
The optimal tax rate is obtained by setting \(\frac{dW}{dt_e} = 0\), is given by
For \(\alpha > \frac{3\lambda \gamma }{b(1 + 4\lambda \gamma )}\), optimal tax rate is positive, \(t_e^{**} > 0\).
The effect \(\alpha \) and \(\gamma \) on opitmal tax rate is given by
Optimal tax rate under absence of green network effect with environmental Regulation
Using equation (22) and substituting the equilibrium values of \(s_H\), \(s_L\) and \(\theta _2 = q_L\) from equation (14, 15) we get 
The optimum is achieved with a small positive emission tax  \(t_e > 0\) if \(328\lambda ^2(2\gamma + \omega )^2 + 524\lambda (2\gamma + \omega ) > 162\lambda \gamma + 209\) which holds for \(\lambda (2\gamma +\omega )>0.33\).
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Grover, C., Bansal, S. Effect of green network and emission tax on consumer choice under discrete continuous framework. Environ Econ Policy Stud (2021). https://doi.org/10.1007/s1001802100312y
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Keywords
 Conflicting quality dimensions
 Discretecontinuous
 Emission tax
 Green network effect
 Vertical differentiation