The many traps of green technology promotion

Abstract

In this paper, we analyze the relationship between technological greening, eco-efficiency and no-regret strategies. By using a simple theoretical model we evaluate the effects of technological greening on creation value, pollution level, and eco-efficiency. We show three contrasting effects of technological greening. First, technological greening may increase the pollution of a firm, and also of the whole industry. Second, the indicator of eco-efficiency can be misleading because it may improve in situations where pollution increases and/or profit decreases after technological greening. Third, technological greening that induces an improvement of the eco-efficiency indicator does not necessarily lead to a no-regret strategy. As a result, the indicator should not be used for decision-making. These are the many traps of technology greening promotion.

Appendices

Appendix 1: Proof of Lemma 1

The effect of technological greening on profit is given by the first derivative of the profit level (2) with respect to the technological parameter b _i . Let us define the following:

$$ \begin{aligned} \varphi_i^{*}(b,t)=\frac{\partial \pi_{i}^{*}}{\partial b_i} -ab_i &= \frac{-t^{2}+b_ipt-2ab_i^{4}}{2b_i^{3}}=0\\ \Leftrightarrow \quad \underline{t}_i &= \frac{1}{2}(b_ip-\sqrt{b_i^{2}p^{2}-8ab_i^{4}})\\ \hbox {and} \quad \bar{t}_i &= \frac{1}{2}(b_ip+\sqrt{b_i^{2}p^{2}-8ab_i^{4}}) \end{aligned} $$

This $\varphi_i$ function allows us to define a frontier on which the effect of technological greening on the profit level is zero, $\varphi_i^{*}(b_i,t)=0.$ There exist cases where real roots of $\varphi_i^{*}(b_i,t)=0$ do not exist. The very existence of real roots for $\varphi_i^{*}(b_i,t)=0,$ denoted by $\underline{t}_i$ and $\bar{t}_i,$ relies on the following assumption:

Assumption

We suppose a condition on a for which an environmental technological amelioration allow an improvement of the profit level. Otherwise $\underline{t}_i$ and $\bar{t}_i$ do not exist which means an environmental technological amelioration decrease the profit level.

$$ b_i^{2}p^{2}-8ab_i^{4}>0 \quad \Leftrightarrow \quad a<\frac{p^2}{8b_i^2} $$

Both of the roots belong to the set domain, $\bar{t}_i<\tilde{t}$ and $\underline{t}_i<\tilde{t}.$ For b _i  = 1, we have:

$$ \begin{aligned} \bar{t}_1 &< t_1\\ \frac{1}{2}(p+\sqrt{p^{2}-8a}) &< p\\ \underline{t}_1 &< t_1\\ \frac{1}{2}(p-\sqrt{p^{2}-16a}) &< p \end{aligned} $$

Calculating the first and the second derivatives of the function $\underline{t}_i=\frac{1}{2}(b_ip-\sqrt{b_i^{2}p^{2}-8ab_i^{4}}),$ we determine that the curve is increasing and convex to the origin.

$$ \begin{aligned} \frac{\partial \underline{t}_i}{\partial b_i} &= \frac{1}{2}\left(p-\frac{-32ab_i^{3}+2b_ip^{2}}{2\sqrt{-8ab_i^{4}+b_i^{2}p^{2}}}\right)>0\\ \frac{\partial^{2} \underline{t}_i}{\partial b_i^{2}} &= \frac{4b_i^{2}(16a^{2}b_i^{2}-3ap^{2})}{(8ab^{2}-p^{2})\sqrt{b_i^{2}p^{2}-8ab^{4}}}>0 \end{aligned} $$

Calculating the first and the second derivatives of the function $\bar{t}_i=\frac{1}{2}(b_ip+\sqrt{b_i^{2}p^{2}-8ab_i^{4}}),$ we determine that the curve is decreasing and concave to the origin.

$$ \begin{aligned} \frac{\partial \bar{t}_i}{\partial b_i} &= \frac{1}{2}\left(p+\frac{-32ab_i^{3}+2b_ip^{2}}{2\sqrt{-8ab_i^{4}+b_i^{2}p^{2}}}\right)<0\\ \frac{\partial^{2} \bar{t}_i}{\partial b_i^{2}} &= - \frac{4b_i^{2}(16a^{2}b_i^{2}-3ap^{2})}{(8ab^{2}-p^{2})\sqrt{b_i^{2}p^{2}-8ab^{4}}}<0 \end{aligned} $$

Appendix 2: Proof of Lemma 2

The effect of technological greening on pollution is given by the first derivative of the emissions (1) with respect to the technological parameter b _i which is given by:

$$ \frac{\partial e_{i}^{*}}{\partial b_i}=\psi_i^{*}(b_i,t) = \frac{2t-b_ip}{2b_i^{3}}=0 \Leftrightarrow \quad \tilde{t} = \frac{b_ip}{2}=0 $$

The function $\tilde{t}_i$ is linear and increasing in b _i .

2.1 Location of the frontier within the set domain

A frontier $\tilde{t}_i=\frac{b_ip}{2}$ is defined such that the effects of technological greening on pollution level is null: ψ ^* _i (b _i ,t) = 0. Under our assumption this function of iso-emissions ($\tilde{t}_i=\frac{b_ip}{2}$) belongs to the set domain $\tilde{t}_1<t_1 \quad \Leftrightarrow \quad p/2<p.$ The function $\tilde{t}_i=b_ip/2$ is located above the frontier $\underline{t}_i=\frac{1}{2}(b_ip-\sqrt{b_i^{2}p^{2}-8ab_i^{4}})$ for b ₁:

$$ \tilde{t}_1> \underline{t}_1 \quad \Leftrightarrow \quad \frac{p}{2}>\frac{1}{2}(p-\sqrt{p^{2}-8a}) \Rightarrow 0>-\sqrt{p^2-8a} $$

2.2 Crossing point

The function $\tilde{t}(a, b_i, p)$ crosses the roots $\bar{t}$ and $\underline{t}$ at a particular value of b _i such that $b_i=\frac{p}{\sqrt{8a}}:$

$$ \begin{aligned} \bar{t}(a, b_i, p)=\tilde{t}(a, b_i, p) &\Leftrightarrow \frac{1}{2}(b_ip+\sqrt{b_i^2p^2-8ab_i^4})=\frac{b_ip}{2} \Rightarrow b_i=\frac{p}{\sqrt{8a}} \\ \underline{t}(a, b_i, p)=\tilde{t}(a, b_i, p) &\Leftrightarrow \frac{1}{2}(b_ip-\sqrt{b_i^2p^2-8ab_i^4})=\frac{b_ip}{2} \Rightarrow b_i=\frac{p}{\sqrt{8a}} \end{aligned} $$

Appendix 3: Proof of Proposition 2

The effect of technological greening on the eco-efficiency indicator is given by the first derivative of I _i w.r.t. b − i, 

$$ \Uplambda_i^{*}(b_i,t)\equiv \frac{\partial I_i}{\partial b_i} = \frac{(\partial \pi_i^{*}-ab_i)e_i^{*}-\pi_i^{*}(\partial e_i^{*})}{(e_i^*)^2}=\frac{-pt+b_ip^2-4ab_i^3}{2(b_ip-t)}=0 \Leftrightarrow \quad \breve{t} = \frac{b_ip^2-4ab_i^3}{p} $$

The function $\breve{t}$ is increasing and concave in b _i and equals zero for $b_i=p/(2\sqrt{a}).$

3.1 Location of the frontier within the set domain

The frontier $\breve{t}_i=\frac{b_ip^2-4ab_i^3}{p}$ is such that the effect of technological greening on eco-efficiency indicator level is nill, $\Uplambda_i^{*}(b_i,t)=0.$ This function $\breve{t}_i=(b_ip^2-4ab_i^3)/p$ belongs to the set domain

$$ t_1 > \breve{t}_1 \quad \Leftrightarrow \quad p>\frac{p^2-4a}{p} \Rightarrow \quad 0>-4a $$

The function $\breve{t} = (b_ip^2-4ab_i^3)/p$ for b _i  = 1 gives us $\breve{t} = (p^2-4a)/p$ and it is located between $\bar{t}_1$ and $\tilde{t}_1.$

$$ \bar{t}_1 >\breve{t}_1\quad \Leftrightarrow \quad\frac{1}{2}(p+\sqrt{p^2-8a})>\frac{p^2-4a}{p} \Rightarrow \quad p> \sqrt{8} $$

Assuming that a < (p ²)/(8b ² _i ) allows us to set a condition on p for b ₁. It confirms that $\bar{t}_1 >\breve{t}_1: a <(p^2)/(8b_i^2) \quad \Leftrightarrow \quad p>\sqrt{8a}.$ Considering this assumption on a we can also confirm that $\breve{t}_1>\tilde{t}_1: \quad \Leftrightarrow \quad (p^2-4a)/p>p/2 \Leftrightarrow p>\sqrt{8a}.$

3.2 Crossing point

The function $\breve{t}(a, b_i, p)$ crosses $\tilde{t}(a, b_i, p)$ and thus $\bar{t}(a, b_i, p)$ and $\underline{t}(a, b_i, p)$ at $b_i=p/(\sqrt{8a})$ (see Appendix 2) such that

$$ \breve{t}(a, b_i, p)=\tilde{t}(a, b_i, p) \quad \Leftrightarrow \quad \frac{b_ip^2-4ab_i^3}{p}=\frac{b_ip}{2} \Rightarrow b_i=\frac{p}{\sqrt{8a}} $$