Multidimensional Green Product Design

Abstract

This paper studies the impact of environmental policies when firms can adjust product design as they see fit. In particular, it considers cross relationships between product design dimensions. For example, when products are designed to be more durable, this may add production steps and increase pollutant emissions during production. More generally, changes applied to one dimension can affect the cost or environmental performance of other dimensions. In this theoretical model, a firm interacts with consumers and a regulator. Before the production stage, the firm must choose the levels of three design dimensions: (1) energy performance during production, (2) energy performance during use, and (3) durability. Depending on the assumptions, the dimensions are said to be complementary, neutral, or competitive. The regulator can promote greener designs by applying targeted environmental taxes on emissions during production or consumption. The main results shed light on the consequences of modifying public policies. When some design dimensions are competitive, a targeted emission tax can result in environmental burden shifting, with an overall increase in pollution. This paper also explores the social optimum and the development of second-best policies when some policy instruments are imperfect. When the social planner ignores the possibility for firms to adjust the level of durability, firms may use planned obsolescence to mitigate the cost of too stringent environmental policies. Also, under particular conditions, a government would want to regulate and constrain the level of durability.

Appendices

Appendix A: Simulations

We build an economy where we define the following functional forms for the unit production cost and pollution damages:

$$\begin{aligned} c(q_{1},q_{2},\delta ;\tau _{1},\tau _{2})= & {} \frac{aq_{1}^{2}}{2}+\frac{dq_{2}^{2}}{2}+\frac{i\delta ^{2}}{2}+bq_{1}q_{2}+cq_{1}\delta +fq_{2}\delta \\ e_{i}(q_{i})= & {} z_{i}-q_{i}\quad \text { for }i=1,2 \end{aligned}$$

where the signs of b, c and f denote the cost cross relationship between the dimensions. The Hessian matrix becomes:

$$\begin{aligned} \varvec{\mathcal {H}}= & {} \left[ \begin{array}{ccc} -c_{q_{1}q_{1}} &{} -c_{q_{1}q_{2}} &{} -c_{q_{1}\delta } \\ -c_{q_{1}q_{2}} &{} -c_{q_{2}q_{2}} &{} \beta \left( p_{e}+\tau _{2}\right) -c_{q_{2}\delta } \\ -c_{q_{1}\delta } &{} \beta \left( p_{e}+\tau _{2}\right) -c_{q_{2}\delta } &{} -c_{\delta \delta } \end{array} \right] \\= & {} \left[ \begin{array}{ccc} h_{11}<0 &{} h_{12} &{} h_{13} \\ h_{12} &{} h_{22}<0 &{} h_{23} \\ h_{13} &{} h_{23} &{} h_{33}<0 \end{array} \right] =\left[ \begin{array}{ccc} -a &{} -b &{} -c \\ -b &{} -d &{} \beta \left( p_{e}+\tau _{2}\right) -f \\ -c &{} \beta \left( p_{e}+\tau _{2}\right) -f &{} -i \end{array} \right] \end{aligned}$$

Note that optimality conditions for the choice of design, Eqs. (2)–(4) can be rewritten as:

$$\begin{aligned} \widehat{q}_{1}(\delta ;\tau _{1},\tau _{2})= & {} \frac{1}{H_{33}}\left[ d\tau _{1}-b(p_{e}+\tau _{2})+H_{13}\delta \right] \\ \widehat{q}_{2}(\delta ;\tau _{1},\tau _{2})= & {} \frac{1}{H_{33}}\left[ a(p_{e}+\tau _{2})-b\tau _{1}+H_{23}\delta \right] \\ \widehat{\delta }(q_{1},q_{2},\delta ;\tau _{1},\tau _{2})= & {} \beta \left( \frac{aq_{1}^{2}}{2}+\frac{dq_{2}^{2}}{2}-\frac{i\delta ^{2}}{2}+bq_{1}q_{2}+\tau _{1}(z_{1}-q_{1})\right) \\&-\,(i\delta +cq_{1}+fq_{2})=0 \end{aligned}$$

which highlight the role of $H_{13}$ and $H_{23}$ in influencing the impact of $\delta$ on $q_{1}$ and $q_{2}$, respectively.

We assign the parameters the following values:

$$\begin{aligned} a=5;d=i=6;\;b=-1;f=-.5;\text { and }c=4.5 \end{aligned}$$

which means that environmental quality during consumption $q_{2}$ is complementary with both quality during production $q_{1}$ and durability $\delta$, whereas quality during production $q_{1}$ and durability $\delta$ are competitive. Other parameters take the values:

$$\begin{aligned} \beta =.97;\;p_{e}=2.4;z_{1}=1.8;z_{2}=2.5\text { and }\tau _{2}=1. \end{aligned}$$

We obtain interior solutions for $\tau _{1}\in [8.3,9]$. The Hessian matrix is negative definite. In the range of interior solutions, an increase in the targeted tax $\tau _{1}$ favors $q_{1}$ and brings a simultaneous reduction in $q_{2}$ and $\delta$. The overall result is that for lower values of $\tau _{1}$, i.e., for $\tau _{1}\in [8.3,8.7]$, a tax increase increases monetary damages of pollution per functional unit $D_{f}$. This is illustrated in Fig. 1.

Fig. 1

Variation in the three design dimensions $q_{1}$, $q_{2}$ and $\delta$; and the emissions per unit of functionality $D_{f}$ with respect to $\tau _{1}$

Figure 2 shows the results using the following set of parameters:

$$\begin{aligned} a= & {} 1,730;d=6.69;i=.10;\,b=-\,8;f=3.30;\text { and }c=9.49 \\ \beta= & {} .97;\;p_{e}=2.4;z_{1}=1.8;z_{2}=2.5\text { and }\tau _{2}=1, \end{aligned}$$

which assumes that environmental quality during consumption $q_{2}$ is complementary with quality during production $q_{1}$, and that the other cross relationships are competitive. We see that an increase in the targeted tax on damage from production $\tau _{1}$ makes products design more polluting during the production stage, i.e., $q_{1}$ decreases.

Fig. 2

Variation in the three design dimensions $q_{1}$, $q_{2}$ and $\delta$; and the emission per unit of functionality $D_{f}$ with respect to $\tau _{1}$

Appendix B: Second-Best Policies

1.1 B.1 When One of the Policy Instruments is Inappropriate

The optimality conditions are $\left( \text {for }t=1,\ldots ,T\right)$:

$$\begin{aligned} \frac{\partial H_{t}}{\partial \psi _{t}}= & {} 0\Leftrightarrow \beta ^{t}x_{t}\left[ \left( -\frac{\partial c(q_{1,t},q_{2,t},\delta _{t})}{\partial q_{1,t}}+1\right) \right. \frac{d\widehat{q}_{1,t}}{d\psi _{t}} \\&+\,\left( (1+\beta \delta _{t})p_{e}-\frac{\partial c(q_{1,t},q_{2,t},\delta _{t})}{\partial q_{2,t}}+(1+\beta \delta _{t})\right) \frac{d\widehat{q}_{2,t}}{d\psi _{t}} \\&+\,\left. \left( \beta (\alpha -p_{e}e_{2}(q_{2,t}))-\frac{\partial c(q_{1,t},q_{2,t},\delta _{t})}{\partial \delta _{t}}-\beta e_{2}(q_{2,t})\right) \frac{d\widehat{\delta }_{t}}{d\psi _{t}}\right] -\lambda _{t}x_{t}\frac{d\widehat{\delta }_{t}}{d\psi _{t}}=0\\ \frac{\partial H_{t}}{\partial x_{t}}= & {} \lambda _{t-1}-\lambda _{t}\Leftrightarrow \beta ^{t}\left[ p(q_{2,t},\delta _{t})-c(q_{1,t},q_{2,t},\delta _{t})-e_{1}(q_{1,t})-(1+\beta \delta _{t})e_{2}(q_{2,t})\right] \\&-\,\lambda _{t}\delta _{t}-\lambda _{t-1}=0\\ \frac{\partial H_{t}}{\partial \lambda _{t}}= & {} x_{t+1}-x_{t}\quad t=0,1,\ldots ,T-1. \end{aligned}$$

1.2 B.2 When the Government also Regulates the Level of Durability

When $\tau _{2}=\overline{\tau }_{2}$ and the government chooses $\tau _{1}$ , $\delta _{t}$, the firm must take durability as given and Eq. (4) is no longer applicable. The Hessian matrix (6) becomes $\varvec{\mathcal {H}}=\left[ \begin{array}{cc} -c_{q_{1}q_{1}} &{} -c_{q_{1}q_{2}} \\ -c_{q_{1}q_{2}} &{} -c_{q_{2}q_{2}} \end{array} \right] =\left[ \begin{array}{cc} h_{11} &{} h_{12} \\ h_{12} &{} h_{22} \end{array} \right]$ and the full impact of a change in a parameter (7) is

$$\begin{aligned} \left[ \begin{array}{c} d\widehat{q}_{1}/d\phi \\ d\widehat{q}_{2}/d\phi \end{array} \right]= & {} -\varvec{\mathcal {H}}^{-1}\left[ \begin{array}{c} \partial f_{1}/\partial \phi \\ \partial f_{2}/\partial \phi \end{array} \right] =\frac{-1}{\det \varvec{\mathcal {H}}}\left[ \begin{array}{cc} h_{22} &{} -h_{12} \\ -h_{12} &{} h_{11} \end{array} \right] \left[ \begin{array}{c} \partial f_{1}/\partial \phi \\ \partial f_{2}/\partial \phi \end{array} \right] \\ \text {and }\det \varvec{\mathcal {H}}= & {} h_{11}h_{22}-h_{12}^{2}=H_{33}>0. \end{aligned}$$

The optimality conditions are $\left( \text {for }t=1,\ldots ,T\right)$:

$$\begin{aligned} \frac{\partial H_{t}}{\partial \tau _{1}}= & {} 0\Leftrightarrow \left( -\frac{\partial c(q_{1,t},q_{2,t},\delta _{t})}{\partial q_{1,t}}+1\right) \frac{d\widehat{q}_{1,t}}{d\tau _{1}} \\&+\,\left( (1+\beta \delta )p_{e}-\frac{\partial c(q_{1,t},q_{2,t},\delta _{t})}{\partial q_{2,t}}+(1+\beta \delta _{t})\right) \frac{d\widehat{q}_{2,t}}{d\tau _{1}}=0 \\ \frac{\partial H_{t}}{\partial \delta _{t}}= & {} \,\beta ^{t}x_{t}\left[ \left( \beta (\alpha -p_{e}e_{2}(q_{2,t}))-\frac{\partial c(q_{1,t},q_{2,t},\delta _{t})}{\partial \delta _{t}}-\beta e_{2}(q_{2,t})\right) \right] -\lambda _{t}x_{t} \\&+\,\frac{\partial H_{t}}{\partial q_{1,t}}\frac{d\widehat{q}_{1,t}}{d\delta _{t}}+\frac{\partial H_{t}}{\partial q_{2,t}}\frac{d\widehat{q}_{2,t}}{d\delta _{t}}=0\\ \frac{\partial H_{t}}{\partial x_{t}}= & {} \lambda _{t-1}-\lambda _{t}\Leftrightarrow \beta ^{t}\left[ p(q_{2,t},\delta _{t})-c(q_{1,t},q_{2,t},\delta _{t})-e_{1}(q_{1,t})-(1+\beta \delta _{t})e_{2}(q_{2,t})\right] \\&-\,\lambda _{t}\delta -\lambda _{t-1}=0 \\ \frac{\partial H_{t}}{\partial \lambda _{t}}= & {} x_{t+1}-x_{t}\quad t=0,1,\ldots ,T-1. \end{aligned}$$

and we use the following properties:

$$\begin{aligned} \frac{\partial f_{1}}{\partial \delta }= & {} -c_{q_{1}\delta }=h_{13} \\ \frac{\partial f_{2}}{\partial \delta }= & {} \beta (p_{e}+\tau _{2})-c_{q_{2}\delta }=h_{23} \\ \frac{d\widehat{q}_{1}}{d\delta }= & {} \frac{-1}{\det \varvec{\mathcal {H}}}(h_{22}h_{13}-h_{12}h_{23})=\frac{H_{13}}{H_{33}} \\ \frac{d\widehat{q}_{2}}{d\delta }= & {} \frac{-1}{\det \varvec{\mathcal {H}}}(-h_{12}h_{13}+h_{11}h_{23})=\frac{H_{23}}{H_{33}} \end{aligned}$$