Some implications of environmental regulation on social welfare under learning-by-doing of eco-products

Koji Kotani; Makoto Kakinaka

doi:10.1007/s10018-015-0140-3

Environmental Economics and Policy Studies

January 2017, Volume 19, Issue 1, pp 121–149 | Cite as

Some implications of environmental regulation on social welfare under learning-by-doing of eco-products

Authors
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Koji KotaniEmail author
Makoto Kakinaka

Research Article

First Online: 08 January 2016

Received: 24 September 2015

Accepted: 12 December 2015

Abstract

This paper examines the significance of environmental regulation in an economy where an eco-product supplied by a single producer is differentiated from a conventional product generating negative externalities. We develop two types of the model: one is a static model without learning effect of eco-product planning, and the other is a dynamic model with learning effect. We show that the regulation should be adopted when the marginal cost of the eco-product production is high enough in a static setting. In a dynamic model, however, whether the regulation improves social welfare is dependent not only on current marginal costs of the eco-product, but also on the degree of dynamic learning effect. Particularly, the regulation could improve social welfare when learning effect is either small or large enough, while it could deteriorate social welfare in an intermediate case. Although intuitions tell us that the value of the regulation appears to be monotonically increasing in learning effect, our results suggest that the value possesses a nonmonotone U-shaped feature with respect to learning effect. The optimal decision of the regulation in a dynamic setting could be converse to that of a static setting, providing important policy implications of learning potentials.

Keywords

Eco-product Environmental regulation Product differentiation Learning-by-doing

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Appendix

We first derive some important variables in the case of static pricing of the eco-product. Using Eq. (3), we obtain \(x_\mathrm{M}(\phi ,c)= \frac{1-c-\gamma (1-\tau (\phi ))}{2(1-\gamma ^{2})}>0\), \(y_\mathrm{M}(\phi ,c)= \frac{(1-\tau (\phi ))(2-\gamma ^{2})-\gamma (1-c)}{2(1-\gamma ^{2})}>0\), and \(\pi _\mathrm{M}(\phi ,c)=\frac{[1-c-\gamma (1-\tau (\phi ))]^{2}}{4(1-\gamma ^{2})}\). Moreover, we obtain \(E_\mathrm{M}(\phi ,c)=\Gamma (\varepsilon y_\mathrm{M}(0,c))\) if \(\phi =0\) and \(E_\mathrm{M}(\phi ,c)=0\) if \(\phi =1\).

Claim 1

For any \(\phi \in \{0,1\}\), technological progress in the production of the eco-product (1) decreases the optimal eco-product price, (2) increases the output of the eco-product, (3) decreases the output of the conventional product and the negative externalities, and (4) increases consumer surplus as well as the profit for the single producer.

Proof of Claim 1

Differentiating \(p_\mathrm{M}\), \(x_\mathrm{M}\) and \(y_\mathrm{M}\) with respect to c yields \(\frac{\partial p_\mathrm{M}}{\partial c}=\frac{1}{2}>0\); \(\frac{\partial x_\mathrm{M}}{\partial c}=-\frac{1}{2(1-\gamma ^{2})}<0\); and \(\frac{\partial y_\mathrm{M}}{\partial c}=\frac{\gamma }{2(1-\gamma ^{2})}>0\), which are the desired results in (1), (2) and (3). Noticing that \(\frac{\partial S}{\partial p}=(u_{x}-p)\frac{\partial x^{*}}{\partial p}+(u_{y}-\tau (\phi ))\frac{\partial y^{*}}{\partial p}-x^{*}=-x^{*}<0\) and that \(x_\mathrm{M}(\phi ,c)>0\) or \(1-c-\gamma (1-\tau (\phi ))>0\), we obtain \(\frac{\partial S_\mathrm{M}}{\partial c}=\frac{\partial S}{\partial p}\frac{\partial p_\mathrm{M}}{\partial c}=-x_\mathrm{M}(\phi ,c)\frac{\partial p_\mathrm{M}}{\partial c}<0\) and \(\frac{\partial \pi _\mathrm{M}}{\partial c}=-\frac{1-c-\gamma (1-\tau (\phi ))}{2(1-\gamma ^{2})}<0\). This is a desired result in (4). \(\square\)

Proof 1

Let \(\tilde{p}(\xi ,c)\equiv p_\mathrm{M}(1,c:\xi )\), \(\tilde{x}(\xi ,c)\equiv x_\mathrm{M}(1,c:\xi )\), \(\tilde{y}(\xi ,c)\equiv y_\mathrm{M}(1,c:\xi )\), \(\tilde{S}(\xi ,c)\equiv S_\mathrm{M}(1,c:\xi )\), and \(\tilde{\pi }(\xi ,c)\equiv \pi _\mathrm{M}(1,c:\xi )\). Notice that \(p_\mathrm{M}(0,c)=\tilde{p}(0,c)\), \(x_\mathrm{M}(0,c)=\tilde{x}(0,c)\), \(y_\mathrm{M}(0,c)=\tilde{y}(0,c)\), \(S_\mathrm{M}(0,c)=\tilde{S}(0,c)\), and \(\pi _\mathrm{M}(0,c)=\tilde{\pi }(0,c)\). Since \(\frac{\partial \tilde{p}}{\partial \xi }=\frac{\gamma }{2}>0\), it must hold that \(p_\mathrm{M}(0,c)=\tilde{p}(0,c)<\tilde{p}(\xi ,c)=p_\mathrm{M}(1,c:\xi )\) for any \(\xi >0\), which is the desired result in (1). Similarly, since \(\frac{\partial \tilde{x}}{\partial \xi }=\frac{\gamma }{2(1-\gamma ^{2})}>0\) and \(\frac{\partial \tilde{y}}{\partial \xi }=-\frac{2-\gamma ^{2}}{2(1-\gamma ^{2})}<0\), it must hold that \(x_\mathrm{M}(0,c)=\tilde{x}(0,c)<\tilde{x}(\xi ,c)=x_\mathrm{M}(1,c:\xi )\) and \(y_\mathrm{M}(0,c)=\tilde{y}(0,c)>\tilde{y}(\xi ,c)=y_\mathrm{M}(1,c:\xi )\) for any \(\xi >0\), which are the desired results in (2). Moreover, notice that \(\frac{\partial \tilde{S}}{\partial \xi }=[u_{x}-\tilde{p}]\frac{\partial \tilde{x}}{\partial \xi }+[u_{y}-(\underline{\tau }+\xi )]\frac{\partial \tilde{y}}{\partial \xi }-\tilde{x}\frac{\partial \tilde{p}}{\partial \xi }-\tilde{y}=-\tilde{x}\frac{\partial \tilde{p}}{\partial \xi }-\tilde{y}<0\). Thus, it must hold that \(S_\mathrm{M}(0,c)=\tilde{S}(0,c)>\tilde{S}(\xi ,c)=S_\mathrm{M}(1,c:\xi )\) for any \(\xi >0\), which is the desired result in the first part of (3). Since \(\frac{\partial \tilde{\pi }}{\partial \xi }=\frac{\gamma [1-c-\gamma (1-\underline{\tau }-\xi )]}{2(1-\gamma ^{2})}>0\), it must hold that \(\pi _\mathrm{M}(0,c)=\tilde{\pi }(0,c)<\tilde{\pi }(\xi ,c)=\pi _\mathrm{M}(1,c:\xi )\) for any \(\xi >0\), which is the desired result in the second part of (3). \(\square\)

Proof 2

Notice \(\Delta T_\mathrm{M}(c)=T(p_\mathrm{M}(1,c),1,c)-T(p_\mathrm{M}(0,c),0,c)=\Gamma (\varepsilon y_\mathrm{M}(0,c))+\Phi (\xi ,c)\) where \(\Gamma (\varepsilon y_\mathrm{M}(0,c))=\Gamma (\varepsilon \frac{(1-\underline{\tau })(2-\gamma ^{2})-\gamma (1-c)}{2(1-\gamma ^{2})})\) and \(\Phi (\xi ,c)=(S_\mathrm{M}(1,c)+\pi _\mathrm{M}(1,c))-(S_\mathrm{M}(0,c)+\pi _\mathrm{M}(0,c))\). \(\Gamma (\varepsilon y_\mathrm{M}(0,c))\) is increasing and convex in c for any \(\varepsilon\), because \(\frac{\partial \Gamma }{\partial c}=\Gamma '(\varepsilon y_\mathrm{M})\varepsilon \frac{\partial y_\mathrm{M}}{\partial c}>0\) and \(\frac{\partial ^{2}\Gamma }{\partial c^{2}}=\varepsilon [\Gamma ''(\varepsilon y_\mathrm{M})\varepsilon \left( \frac{\partial y_\mathrm{M}}{\partial c}\right) ^{2}+\Gamma '(\varepsilon y_\mathrm{M})\frac{\partial ^{2} y_\mathrm{M}}{\partial c^{2}}]>0\). Moreover, \(\Phi (\xi ,c)\) is strictly decreasing in c with linearity since \(\frac{\partial \Phi (\xi ,c)}{\partial c}=-\frac{3\gamma \xi }{4(1-\gamma ^{2})}<0\). Taking into account that \(\Delta T_\mathrm{M}(0)<0\) and \(\Delta T_\mathrm{M}(1)>0\) and that \(\Gamma\) is increasing and convex, it must hold that there exists a unique critical value \(c_\mathrm{M}>0\) such that \(\Phi (\xi ,c)>-\Gamma (\varepsilon y_\mathrm{M}(0,c))\) or \(T(p_\mathrm{M}(1,c),1,c)>T(p_\mathrm{M}(0,c),0,c)\) if \(c>c_\mathrm{M}\) and \(\Phi (\xi ,c)<-\Gamma (\varepsilon y_\mathrm{M}(0,c))\) or \(T(p_\mathrm{M}(1,c),1,c)<T(p_\mathrm{M}(0,c),0,c)\) if \(c<c_\mathrm{M}\). \(\square\)

Claim 2

The critical value \(c_\mathrm{M}\) is decreasing in the pollution level per unit of the output of the conventional product, \(\varepsilon\). Furthermore, the critical value \(c_\mathrm{M}\) is increasing in the effect of the regulation on the marginal cost of the conventional product, \(\xi\).

Proof of Claim 2

Notice that \(\Delta T_\mathrm{M}(c)=\Phi (\xi ,c)+\Gamma (\varepsilon y_\mathrm{M}(0,c))=0\) is increasing in c in the neighborhood of \(c=c_\mathrm{M}(\xi ,\varepsilon )\). This implies that at \(c=c_\mathrm{M}(\xi ,\varepsilon )\), \(\Gamma '(\varepsilon y_\mathrm{M})\varepsilon \frac{\partial y_\mathrm{M}}{\partial c}+\frac{\partial \Phi }{\partial c}>0\). Differentiating \(\Delta T_\mathrm{M}(c_{M}(\xi ,\varepsilon ))=\Phi (\xi ,c_\mathrm{M}(\xi ,\varepsilon ))+\Gamma (\varepsilon y_\mathrm{M}(0,c_\mathrm{M}(\xi ,\varepsilon )))=0\) with respect to \(\varepsilon\) yields \(\frac{\partial c_\mathrm{M}(\xi ,\varepsilon )}{\partial \varepsilon }=-\Gamma '(\varepsilon y_\mathrm{M})y_{M}[\Gamma '(\varepsilon y_\mathrm{M})\varepsilon \frac{\partial y_\mathrm{M}}{\partial c}+\frac{\partial \Phi }{\partial c}]^{-1}<0\) since \(\Gamma '(\varepsilon y_\mathrm{M})\varepsilon \frac{\partial y_\mathrm{M}}{\partial c}+\frac{\partial \Phi }{\partial c}>0\) and \(\Gamma '>0\). Thus, \(c_\mathrm{M}(\xi ,\varepsilon )\) is decreasing in \(\varepsilon\). Furthermore, differentiating \(\Delta T_\mathrm{M}(c_{M}(\xi ,\varepsilon ))=0\) with respect to \(\xi\) yields that \(\frac{\partial c_\mathrm{M}(\xi ,\varepsilon )}{\partial \xi }=-\frac{\partial \Phi }{\partial \xi }\left[ \Gamma '(\varepsilon y_\mathrm{M})\varepsilon \frac{\partial y_\mathrm{M}}{\partial c}+\frac{\partial \Phi }{\partial c}\right] ^{-1}> 0\;\Leftrightarrow \;\frac{\partial \Phi }{\partial \xi }< 0\). Thus, \(c_\mathrm{M}(\xi ,\varepsilon )\) is increasing in \(\xi\). \(\square\)

Claim 3

For any \(\phi \in \{0,1\}\), there exists a unique steady state \((\bar{c}(\phi ),\bar{p}(\phi ))\). Furthermore, the unique steady state is stable if the degree of product differentiation between the eco-product and the conventional product is small enough so that \(\text {det}(M)<0\) or \(\gamma <\hat{\gamma }\), where \(\hat{\gamma }\equiv (1-\frac{r+2\lambda \delta }{2\delta (r+\lambda \delta )})^{1/2}\).

Proof of Claim 3

Since the system (4) is a linear differential system, the desired result in the first part is directly obtained. The proof of the second part follows Kamien and Schwartz (2012). The characteristic equation associated with Eq. (4) is \(k^{2}-rk-\lambda \delta (r+\lambda \delta )+\frac{(r+2\lambda \delta )\lambda }{2(1-\gamma ^{2})}=0\) with roots \(k_{1},k_{2}=\frac{r}{2}\pm \frac{1}{2}[(r+2\lambda \delta )^{2}-\frac{2\lambda (r+2\lambda \delta )}{1-\gamma ^{2}}]^{1/2}\equiv \frac{r}{2}\pm \frac{1}{2}D^{1/2}\). Then, the solution has the form of \(c(t)=A\mathrm{e}^{k_{1}t}+B\mathrm{e}^{k_{2}t}\) for \(k_{1}\), \(k_{2}\) real and distinct, or \(c(t)=(A+Bt)\mathrm{e}^{rt/2}\) if \(k_{1}=k_{2}=r/2\), or \(c(t)=\mathrm{e}^{rt/2}(A\cos pt+B\sin pt)\), where \(p=(-D)^{1/2}\), if p is real. If the roots are real, the larger root must be positive. The smaller root may be either positive or negative. It will be negative if \(r<D^{1/2}\); that is if \(\delta (r+\lambda \delta )>\frac{r+2\lambda \delta }{2(1-\gamma ^{2})}\) or \(\gamma ^{2}<1-\frac{r+2\lambda \delta }{2\delta (r+\lambda \delta )}\equiv \hat{\gamma }^{2}\), the roots are real and of opposite signs. Let \(k_{1}>0>k_{2}\) if \(\gamma ^{2}<\hat{\gamma }^{2}\) holds. Then, c(t) will converge to zero provided we take \(A=0\). Thus, if \(\gamma ^{2}<\hat{\gamma }^{2}\) holds, then the roots must be real and the steady state must satisfy the conditions to be a saddlepoint. On the other hand, the roots will be real and nonnegative if \(D\ge 0\) and \(\gamma ^{2}<\hat{\gamma }^{2}\) fails. As long as the roots are both nonnegative, c(t) cannot converge to the steady state. It will move away from it, unless the initial position happens to be the steady state. Finally, the roots will be complex if \(D<0\). Note that since the real part of the complex roots is positive \((=r/2)\), the path moves away from the steady state so that the steady state is unstable. In sum, a solution to the system (4) can converge only if \(\gamma ^{2}<\hat{\gamma }^{2}\) holds. In other cases, all paths diverge. \(\square\)

Claim 4

For any \(\phi\), if \(\lambda >0\) and \(\gamma <\hat{\gamma }\), in the steady state, an increase in \(\lambda\) (1) reduces the marginal cost and the price of the eco-product; (2) raises the output of the eco-product; (3) reduces the output of the conventional product and the negative externalities; and (4) increases consumer surplus.

Proof of Claim 4

By Eq. (6), we obtain \(x_\mathrm{M}(\phi ,\bar{c}(\phi ))=\sigma (\bar{c}(\phi ))[1-\frac{\lambda }{2(r+\lambda \delta )(1-\gamma ^{2})}]\). We first show \(\frac{\partial \bar{c}}{\partial \lambda }<0\). Differentiating this with respect to \(\lambda\) yields \(\frac{\partial \bar{c}(\phi )}{\partial \lambda }=-\frac{r\sigma (\bar{c})}{2(r+\lambda \delta )^{2}(1-\gamma ^{2})}[\frac{\partial x_\mathrm{M}}{\partial c}-\frac{\partial \sigma }{\partial c}(1-\frac{\lambda }{2(r+\lambda \delta )(1-\gamma ^{2})})]^{-1}\). Notice that since \(\gamma <\hat{\gamma }\), we obtain \(\frac{\partial x_\mathrm{M}}{\partial c}-\frac{\partial \sigma }{\partial c}(1-\frac{\lambda }{2(r+\lambda \delta )(1-\gamma ^{2})})=-\frac{\delta }{1-\gamma ^{2}} [\gamma ^{2}-(1-\frac{r+2\lambda \delta }{2\delta (r+\lambda \delta )})]>0\). Thus, we obtain \(\frac{\partial \bar{c}}{\partial \lambda }<0\), which is the desired result in the first part of (1). Note that \(\bar{p}(\phi )=p_\mathrm{M}(\phi ,\bar{c}(\phi ))-\frac{\lambda \sigma (\bar{c}(\phi ))}{2(r+\lambda \delta )}\). Differentiating this with respect to \(\lambda\) yields \(\frac{\partial \bar{p}(\phi )}{\partial \lambda }=\frac{\partial p_\mathrm{M}}{\partial c}\frac{\partial \bar{c}}{\partial \lambda }-\frac{1}{2}[\frac{r\sigma (\bar{c}(\phi ))}{(r+\lambda \delta )^{2}}-\frac{\lambda \delta }{r+\lambda \delta }\frac{\partial \bar{c}}{\partial \lambda }]\). Since \(\frac{\partial p_\mathrm{M}}{\partial c}>0\) and \(\frac{\partial \bar{c}}{\partial \lambda }<0\), it must hold that \(\frac{\partial \bar{p}(\phi )}{\partial \lambda }<0\), which is the desired result in the second part of (1). We then show that \(\frac{\partial \bar{x}}{\partial \lambda }>0\) and \(\frac{\partial \bar{y}}{\partial \lambda }<0\), which are the desired results in (2) and (3). By Eq. (1), the outputs of the eco-product and the conventional product in the steady state are given by \(\bar{x}(\phi )=x^{*}(\bar{p}(\phi ),\phi )\) and \(\bar{y}(\phi )=y^{*}(\bar{p}(\phi ),\phi )\), which implies that \(\frac{\partial \bar{x}(\phi )}{\partial \lambda }=\frac{\partial x^{*}}{\partial p}\frac{\partial \bar{p}(\phi )}{\partial \lambda }=-\frac{1}{1-\gamma ^{2}}\frac{\partial \bar{p}(\phi )}{\partial \lambda }>0\) and \(\frac{\partial \bar{y}(\phi )}{\partial \lambda }=\frac{\partial y^{*}}{\partial p}\frac{\partial \bar{p}(\phi )}{\partial \lambda }=\frac{\gamma }{1-\gamma ^{2}}\frac{\partial \bar{p}(\phi )}{\partial \lambda }<0\). We next show that \(\frac{\partial \bar{S}(\phi )}{\partial \lambda }>0\), which is the desired result in (4). Using \(\bar{S}(\phi )=S(\bar{p}(\phi ),\phi )\), \(\frac{\partial \bar{p}}{\partial \lambda }<0\) and \(\frac{\partial S}{\partial p}=-\bar{x}(\phi )\), we obtain \(\frac{\partial \bar{S}(\phi )}{\partial \lambda }=\frac{\partial S}{\partial p}\frac{\partial \bar{p}}{\partial \lambda }=-\bar{x}(\phi )\frac{\partial \bar{p}}{\partial \lambda }>0\). \(\square\)

Proof 3

Since \(\hat{\gamma }>\gamma\), it must hold that \(\Delta \bar{c}\equiv \bar{c}(0)-\bar{c}(1)>0\), which is the desired result of the first part. Moreover, since \(\frac{\partial \hat{\gamma }}{\partial \lambda }<0\), it must hold that \(\frac{\partial (\Delta \bar{c})}{\partial \lambda }>0\), which is the desired result of the second part. \(\square\)

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Authors and Affiliations

Koji Kotani
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Email authorView author's OrcID profile
Makoto Kakinaka
- 2

1.School of Economics and ManagementKochi University of TechnologyKochi-shiJapan
2.Graduate School of International RelationsInternational University of JapanMinami-UonumaJapan

Some implications of environmental regulation on social welfare under learning-by-doing of eco-products

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