Green certification, heterogeneous producers, and green consumers: a welfare analysis of environmental regulations

Abstract

We develop a vertical differentiation model to analyze welfare implications of environmental policies in a competitive market with production and consumption heterogeneity. Consumers with heterogeneous preferences choose between non-green and certified green products, while producers with heterogeneous production costs decide whether to engage in green production. In order for green products to be recognized by consumers, producers must join a green club. Key findings are summarized as follows. (i) The number of green producers, environmental standard, and overall welfare under the market solution are all socially sub-optimal. (ii) The introduction of a subsidy policy for greener production and standards is shown to increase social welfare, but is not Pareto optimal. (iii) A dual policy, which combines abatement subsidizes for a greener production standard and a tax charge for green certification, is shown to be the Pareto-optimal outcome.

Appendices

Appendix A-1

Given that the equilibrium number of green producers in the market is: $n^{{ GC}}=\frac{\tau }{(\tau +2\sqrt{\gamma \varepsilon })}$, we take the derivative of $n^{{ GC}}$ with respect to $\tau ,\varepsilon$, and $\gamma$ to obtain the following:

$$\begin{aligned}&\frac{\partial n^{{ GC}}}{\partial \tau }=\frac{2[\tau ^{2}\sqrt{\gamma \varepsilon }-4\gamma \varepsilon (\tau +\sqrt{\gamma \varepsilon })]}{(\tau ^{2}-4\gamma \varepsilon )^{2}}>0,\\&\frac{\partial n^{{ GC}}}{\partial \varepsilon }=\frac{-\tau \gamma }{\sqrt{\gamma \varepsilon }(\tau +2\sqrt{\gamma \varepsilon })^{2}}<0,\\&\frac{\partial n^{{ GC}}}{\partial \gamma }=\frac{-\tau \varepsilon }{\sqrt{\gamma \varepsilon }(\tau +2\sqrt{\gamma \varepsilon })^{2}}<0. \end{aligned}$$

1.1 Appendix A-2

Given that the equilibrium value of the green product premium is: $P_e^{{ GC}} =\frac{2\tau \gamma }{\tau +2\sqrt{\gamma \varepsilon }}$, we take the derivative of $P_e^{{ GC}}$ with respect to $\tau ,\varepsilon$, and $\gamma$ to obtain the following:

$$\begin{aligned}&\frac{\partial P_e^{{ GC}} }{\partial \tau }=\frac{4\gamma (\tau ^{2}\sqrt{\gamma \varepsilon }-4\tau \gamma \varepsilon +4\gamma \varepsilon \sqrt{\gamma \varepsilon })}{(\tau ^{2}-4\gamma \varepsilon )^{2}}>0,\\&\frac{\partial P_e^{{ GC}} }{\partial \varepsilon }=\frac{-2\tau \gamma ^{2}}{\sqrt{\gamma \varepsilon }(\tau +2\sqrt{\gamma \varepsilon })^{2}}<0,\\&\frac{\partial P_e^{{ GC}} }{\partial \gamma }=\frac{2\tau (\tau +\sqrt{\gamma \varepsilon })}{(\tau +2\sqrt{\gamma \varepsilon })^{2}} >0. \end{aligned}$$

Appendix A-3

1.1 Consumer surplus

According to the preferences of heterogeneous consumers as specified in (1), we have

$$\begin{aligned} { CS}=\int _0^x {[v+(1-x) \tau \theta +\beta \psi -(P+P_e )]dx} +\int _x^1 {(v+\beta \psi -P)dx} . \end{aligned}$$

Note that $\beta \psi$ is the external benefit to the society from the green product’s environmental friendliness or abatement, where $\psi =\theta n$ and n is the equilibrium quantity of the green product sold in the market. We then have

$$\begin{aligned} { CS}=\int _0^x {[v+(1-x) \tau \theta +\beta \theta n-(P+P_e )]dx} +\int _x^1 {(v+\beta \theta n-P)dx} , \end{aligned}$$

which is re-written as

$$\begin{aligned} { CS}= & {} \int _0^x {[(1-x) \tau \theta -P_e ]dx} +\int _0^x {(v+\beta \theta n-P)dx} +\int _x^1 {(v+\beta \theta n-P)dx} \\= & {} \left[ {\tau \theta x-\frac{\tau \theta x^{2}}{2}-P_e x} \right] _0^x +\left[ {vx+\beta \theta nx-Px} \right] _0^1 \\= & {} \left[ {\frac{\theta \tau x(2-x) }{2}-P_e x} \right] +(v+\beta \theta n-P). \end{aligned}$$

Competitive market for the non-green product implies that the equilibrium price for the good is equal to its price, that is, $P=v$. In addition, the equilibrium quantity of the green product sold (n) is equal to the number of green consumers (x). It follows that

$$\begin{aligned} { CS}=\beta \theta n+\frac{\theta \tau x(2-x) }{2}-P_e x, \end{aligned}$$

where $\beta \theta n$ is public benefit from the green product, $[{\theta \tau x(2-x)]}/2$ is private benefit to green consumers, and $P_e x$ is the amount of premium to green producers.

Appendix A-4

1.1 Producer surplus

According to the profit functions of green and non-green producers as specified in (3), we have

$$\begin{aligned} { PS}=\int _0^y {(P+P_e -y\varepsilon \theta ^{2}-c-\gamma y)dy} +\int _y^1 {(P-c)dy} , \end{aligned}$$

which is re-written as

$$\begin{aligned} { PS}= & {} \int _0^y {(P_e -y\varepsilon \theta ^{2}-\gamma y)dy} +(P-c) \\= & {} \left[ {P_e -\frac{\varepsilon \theta ^{2}y^{2}}{2}-\frac{\gamma y^{2}}{2}} \right] _0^y +(P-c) \\= & {} P_e y-\frac{(\varepsilon \theta ^{2}+\gamma )y^{2}}{2}+(P-c). \end{aligned}$$

Competitive market for the green product implies that the equilibrium price for the good is equal to the cost of production for the marginal producer, that is, $P=c$. We thus have

$$\begin{aligned} { PS}=P_e y-\frac{(\varepsilon \theta ^{2}+\gamma )y^{2}}{2}. \end{aligned}$$

where $P_e y$ is green price premium, and ${(\varepsilon \theta ^{2}+\gamma )y^{2}}/2$ is green cost.

Appendix A-5

Given that the equilibrium level of social welfare is:

$$\begin{aligned} SW^{{ GC}}=\frac{\tau \sqrt{\gamma \varepsilon }(2\beta +\tau )}{2\varepsilon (\tau +2\sqrt{\gamma \varepsilon })}, \end{aligned}$$

we take the derivative of $SW^{{ GC}}$ with respect to $\tau ,\varepsilon$ ,and $\gamma$ to obtain the following:

$$\begin{aligned}&\frac{\partial { SW}^{{ GC}}}{\partial \tau }=\frac{\left( {\tau -2\sqrt{\gamma \varepsilon }} \right) ^{2}(\tau ^{2}\sqrt{\gamma \varepsilon }+4\beta \gamma \varepsilon +4\tau \gamma \varepsilon )}{2\varepsilon (\tau ^{2}-4\gamma \varepsilon )^{2}}>0,\\&\frac{\partial { SW}^{{ GC}}}{\partial \varepsilon }=-\frac{\tau \sqrt{\gamma \varepsilon }(2\beta +\tau )\left[ \tau ^{3}+4\gamma \varepsilon (4\sqrt{\gamma \varepsilon }-3\tau )\right] }{4\varepsilon ^{2}(\tau ^{2}-4\gamma \varepsilon )^{2}}<0,\\&\frac{\partial { SW}^{{ GC}}}{\partial \gamma }=\frac{\tau ^{2}(2\beta +\tau )(\tau ^{2}+4\gamma \varepsilon -4\tau \sqrt{\gamma \varepsilon })\sqrt{\gamma \varepsilon }}{4\gamma \varepsilon (\tau ^{2}-4\gamma \varepsilon )^{2}}>0. \end{aligned}$$