Green products, market structure, and welfare

Abstract

This paper examines the welfare consequences of private provision of green goods in output markets with product differentiation. In our setting, consumers can be prone to engage in the consumption of green goods (e.g., from eco-labels or due to some forms of altruism), and firm entry is endogenous. Our analysis shows that firms underinvest in environmentally cleaner products in situations where the social damage from polluting emissions is sufficiently large. However, beyond those situations, we find that firms can underinvest or overinvest depending on effects mediated through firm entry because entry raises output and thus increases consumer participation in the market, but it also reduces private incentives to provide consumers with cleaner products.

Appendix

1.1 Welfare benchmark

Here, we derive the welfare benchmark in our analysis. The welfare function is written as $W=CS+\Pi -D$, where CS denotes consumer surplus, $\Pi$ industry profits, and D the aggregate damage due to the environmental externality from emissions.

At a first-best configuration,

$$\begin{aligned} CS&= \gamma ^{2}\left( \int \limits _{0}^{1/2}\left( v-\theta e-x-p\right) dx+\int \limits _{1/2}^{1}\left( v-\theta e-(1-x)-p\right) dx\right) \\&\quad +2\gamma (1-\gamma )\int \limits _{0}^{1}\left( v-\theta e-x-p\right) dx \\&= \gamma ^{2}\left( v-\theta e-\frac{1}{4}-p\right) +2\gamma (1-\gamma )\left( v-\theta e-\frac{1}{2}-p\right) ,\\ \Pi&= 2\gamma \left( \frac{2-\gamma }{2}(p-c)-f-\beta (e)\right) =(2-\gamma )\gamma (p-c)-2\gamma (f+\beta (e)),\\ D&= \lambda E=\lambda (\gamma ^{2}+2\gamma (1-\gamma ))e=\lambda (2-\gamma )\gamma e, \end{aligned}$$

from where

$$\begin{aligned} W=\gamma ^{2}\left( v-\theta _{w}e-\frac{1}{4}-c\right) +2\gamma (1-\gamma )\left( v-\theta _{w}e-\frac{1}{2}-c\right) -2\gamma (f+\beta (e)), \end{aligned}$$

with $\theta _{w}\equiv \theta +\lambda$, which gives W as in equation (1).

1.2 Proofs of the results

Proofs of Propositions 1 and 2

See Granero (2019). $\square$

Proof of Proposition 3

From equations (8)–(9),

$$\begin{aligned}&\frac{\partial ^{2}\pi _{i}}{\partial p_{i}^{2}}=-\gamma<0,\ \ \ \frac{ \partial ^{2}\pi _{i}}{\partial e_{i}^{2}}=-\beta ^{\prime \prime }(e_{i})<0,\\&\frac{\partial ^{2}\pi _{i}}{\partial p_{i}^{2}}\frac{\partial ^{2}\pi _{i}}{ \partial e_{i}^{2}}-\left( \frac{\partial ^{2}\pi _{i}}{\partial p_{i}\partial e_{i}}\right) ^{2}=\gamma \left\{ \beta ^{\prime \prime }(e_{i})-\left( \frac{\theta }{2}\right) ^{2}\gamma \right\} , \end{aligned}$$

where

$$\begin{aligned} \frac{\partial ^{2}\pi _{i}}{\partial p_{i}\partial e_{i}}=\frac{\partial ^{2}\pi _{i}}{\partial e_{i}\partial p_{i}}=-\frac{\gamma \theta }{2}. \end{aligned}$$

Hence, $(\partial ^{2}\pi _{i}/\partial p_{i}^{2})(\partial ^{2}\pi _{i}/\partial e_{i}^{2})-\left( \partial ^{2}\pi _{i}/\partial p_{i}\partial e_{i}\right) ^{2}>0$ iff $\beta ^{\prime \prime }(e_{i})>\gamma \theta ^{2}/4$, which is implied by $\beta ^{\prime \prime }(e_{i})>\theta _{w}^{2}/4$ given that $\gamma \le 1$ and $\theta \le \theta _{w}$. Then, expressions (10) and (11) yield the profit-maximizing solution $(p_{i},e_{i})$, and thus at a symmetric equilibrium $(p_{i},e_{i})=(p^{*},e^{*})$ as given by equations (12) and (13) for $f\in [\underline{f},\widehat{f}]$, i.e., $\gamma \ge \widehat{\gamma }$, provided condition (14) holds. In those circumstances, free entry yields $\gamma ^{*}$ as the solution to the zero profit condition (15). For any $\gamma \ge \widehat{\gamma }$ that condition is equivalent to

$$\begin{aligned} g(\gamma )\equiv \gamma \pi ^{*}(\gamma )=\frac{(2-\gamma )^{2}}{2} -\gamma \beta (e^{*}(\gamma ))-\gamma f=0, \end{aligned}$$

where $e^{*}(\gamma )$ follows from $-\beta ^{\prime }(e^{*})=\theta (2-\gamma )/2$. We have that $g(\gamma )$ is a continuous function, $g^{\prime }(\gamma )<0$ when $\pi ^{*}(\gamma )\ge 0$, $g(\widehat{ \gamma })>0$ for $f<\widehat{f}$, and $g(1)\le 0$. Hence, there exists one solution to $g(\gamma )=0$ given by $\gamma ^{*}\in (\widehat{\gamma } ,1)$. If $f=\widehat{f}$ then $\gamma ^{*}=\widehat{\gamma }$ as given by

$$\begin{aligned} \frac{(v-\theta e^{*}(\widehat{\gamma })-1-c)^{2}}{v-\theta e^{*}( \widehat{\gamma })-c}-\beta (e^{*}(\widehat{\gamma }))=\widehat{f}, \end{aligned}$$

with $e^{*}(\widehat{\gamma })$ such that $-\beta ^{\prime }(e^{*})=\theta (2-\widehat{\gamma })/2$; and if $f\le \underline{f}$ then $\gamma ^{*}=1$.

Now, consider $f>\widehat{f}$, i.e., $\gamma <\widehat{\gamma }$. Here, we need to check that the only symmetric equilibrium involves $p^{*}=v-\theta e^{*}-1$ and $-\beta ^{\prime }(e^{*})=\theta (2-\gamma )/2$. A representative firm i chooses $(p_{i},e_{i})$ to maximize

$$\begin{aligned} \pi _{i}=\left[ \gamma \left( \frac{1}{2}+\frac{\theta (e-e_{i})+p-p_{i}}{2} \right) +(1-\gamma )(v-\theta e_{i}-p_{i})\right] (p_{i}-c)-\beta (e_{i})-f, \end{aligned}$$

subject to $p_{i}+\theta e_{i}\ge v-1$. The first-order conditions for an interior solution can be written as

$$\begin{aligned}&\gamma \left( 1+\theta (e-e_{i})+p-2p_{i}+c\right) +2(1-\gamma )(v-\theta e_{i}-2p_{i}+c)=0,\\&\qquad-\frac{\theta }{2}(2-\gamma )(p_{i}-c)-\beta ^{\prime }(e_{i})=0. \end{aligned}$$

If a symmetric equilibrium exists, then the price is given by

$$\begin{aligned} p(\gamma )=\frac{2(1-\gamma )(v-\theta e(\gamma )+c)+\gamma (1+c)}{4-3\gamma }, \end{aligned}$$

where $e(\gamma )$ is such that

$$\begin{aligned} -\beta ^{\prime }(e)=\frac{\theta }{2}\frac{2-\gamma }{4-3\gamma } (2(1-\gamma )(v-\theta e-c)+\gamma ). \end{aligned}$$

It turns out that $p(0)+\theta e(0)=\frac{1}{2}(v+\theta e(0)+c)<v-1$, and $p^{\prime }(\gamma )<0$. Thus, a contradiction follows. If other firms set $p+\theta e=v-1$ then according to firm i’s first-order conditions its best response is such that $p_{i}+\theta e_{i}=\frac{1}{2}(v+\theta e_{i}+c)<v-1$. Therefore, the only symmetric equilibrium involves $p+\theta e=v-1$, from where $p^{*}=v-\theta e^{*}-1$ and $-\beta ^{\prime }(e^{*})=\theta (2-\gamma )/2$. Then, free entry determines $\gamma ^{*}$ as the solution to the zero profit condition (19) if $f\in [\widehat{f}, \overline{f}]$, and $\gamma ^{*}=0$ if $f\ge \overline{f}$. This shows the result.

Proof of Proposition 4

Consider first $f\in (\underline{f}, \widehat{f})$. Then, $\gamma ^{*}$ is given by the solution to the zero profit condition (15). That condition is equivalent to

$$\begin{aligned} g(\gamma )\equiv \gamma \pi ^{*}(\gamma )=\frac{(2-\gamma )^{2}}{2} -\gamma \beta (e^{*}(\gamma ))-\gamma f=0, \end{aligned}$$

where $e^{*}(\gamma )$ follows from $\beta ^{\prime }(e^{*})=-\theta (2-\gamma )/2$. Hence, $\gamma ^{*}$ is given by the only solution to $g(\gamma )=0$ such that $\gamma ^{*}\in (\widehat{\gamma } ,1)$, and implicit differentiation gives rise to

$$\begin{aligned} \frac{d\gamma ^{*}}{df}=-\frac{\gamma ^{*}}{(2-\gamma ^{*})\left( 1-\frac{\theta ^{2}}{4\beta ^{\prime \prime }(e^{*})}\gamma ^{*}\right) +\beta (e^{*})+f}<0. \end{aligned}$$

Next, consider that $f\in (\widehat{f},\overline{f})$. Then, $\gamma ^{*}$ is given by the solution to the zero profit condition (19), and implicit differentiation yields

$$\begin{aligned} \frac{d\gamma ^{*}}{df}=-\frac{2}{v-\theta e^{*}-1-c}<0, \end{aligned}$$

and if $f\ge \overline{f}$ then $\gamma ^{*}=0$. Thus, $\gamma ^{*}$ is weakly decreasing in f, and it is strictly decreasing in f for $f\in ( \underline{f},\overline{f})$. Since $e^{*}$ is strictly increasing in $\gamma$ for $f\in (\underline{f},\overline{f})$, this implies that $e^{*}$ is weakly decreasing in f, and it is strictly decreasing in f for $f\in (\underline{f},\overline{f})$. Finally, because $\gamma ^{*}$ and $e^{*}$ are strictly decreasing in f for $f\in (\underline{f},\widehat{f })$, it follows that $p^{*}$ is weakly increasing in f, and it is strictly increasing in f for $f\in (\underline{f},\overline{f})$. Thus, the result is shown.

Proof of Proposition 5

With an exogenous number of firms, part (i) follows directly from equations (3), (12) and (13) under $\theta <\theta _{w}$. Next, with an endogenous number of firms it can be seen that there exists a positive threshold $\overline{\alpha }$ such that $\partial W(\gamma ^{*},e^{*})/\partial e<0$ for all $\lambda =\theta _{w}-\theta >\overline{\alpha }$. Hence, there exists no crossing point at which $e^{*}(\gamma ^{*})=e^{w}(\gamma ^{w})$ whenever the difference $\theta _{w}-\theta$ is above $\overline{\alpha }$. In particular, $e^{*}(\gamma ^{*})>e^{w}(\gamma ^{w})$ for all $\lambda >\overline{\alpha }$, so that part (ii.1) holds.

Consider now part (ii.2), so that $\lambda <\overline{\alpha }$. Here, there exists a positive threshold $\underline{\alpha }<\overline{\alpha }$ such that with $\lambda \in (\underline{\alpha },\overline{\alpha })$ we have $\partial W(\gamma ^{*},e^{*})/\partial e\gtreqless 0$ as $f\gtreqless f_{I}^{a}$ for $f\in [\underline{f},\overline{f})$, and $\partial W(\gamma ^{*},e^{*})/\partial e\lesseqgtr 0$ as $f\gtreqless f_{I}^{b}$ for $f\in [\overline{f},\overline{f}_{w}]$. Consequently, in the region where $\widehat{f}\le f\le \overline{f}$ we can use equations (2) and (19) to see that as long as $\lambda <\overline{ \alpha }$ there exists one crossing point at which $e^{w}(\gamma ^{w})=e^{*}(\gamma ^{*})$, where $e^{w}(\gamma ^{w})$ follows from (3), and $e^{*}(\gamma ^{*})$ follows from (13). However, in the region where $\underline{f}\le f\le \widehat{f}$ no crossing point at which $e^{w}(\gamma ^{w})=e^{*}(\gamma ^{*})$ does exist whenever $\lambda >\underline{\alpha }$. Hence, part (ii.2) follows.

Finally, consider part (iii.3), where $\lambda <\underline{\alpha }$. First, consider the region $\widehat{f}\le f\le \overline{f}_{w}$. Making use of (2) and (19), here $\gamma ^{*}$ and $\gamma ^{w}$ cross as long as

$$\begin{aligned} \frac{2-\gamma }{2}\left( v-\theta e^{*}(\gamma )-1-c\right) =\frac{1}{4} \gamma +(1-\gamma )\left( v-\theta e^{w}(\gamma )-\frac{1}{2}-c\right) , \end{aligned}$$

with $e^{*}(\gamma )=e^{w}(\gamma )$ for $\gamma =\gamma ^{*}=\gamma ^{w}$ under $\theta$ arbitrarily close to $\theta _{w}$. This holds whenever

$$\begin{aligned} h_{I}(\gamma )\equiv \gamma \left( v-\theta e^{w}(\gamma )-\frac{1}{2} -c\right) -1=0. \end{aligned}$$

It can be seen that $h_{I}(0)<0$, $h_{I}(1)>0$, and $h_{I}^{\prime }(\gamma )>0$. Hence, in this region there exists one solution $\gamma \in (0,1)$ to $h_{I}(\gamma )=0$. That is, there exists one crossing point at which $\gamma ^{*}=\gamma ^{w}$ when $\theta$ approaches $\theta _{w}$. Denote that crossing point by $\gamma (f_{I}^{a})$.

Next, consider the region $\underline{f}_{w}\le f\le \widehat{f}$. From (2) and (15), here $\gamma ^{*}$ and $\gamma ^{w}$ cross as long as

$$\begin{aligned} \frac{(2-\gamma )^{2}}{2\gamma }=\frac{1}{4}\gamma +(1-\gamma )\left( v-\theta e^{w}(\gamma )-\frac{1}{2}-c\right) , \end{aligned}$$

with $e^{*}(\gamma )=e^{w}(\gamma )$ for $\gamma =\gamma ^{*}=\gamma ^{w}$ under $\theta$ arbitrarily close to $\theta _{w}$. This holds whenever

$$\begin{aligned} h_{II}(\gamma )\equiv 2\gamma (1-\gamma )\left( v-\theta e^{w}(\gamma )- \frac{1}{2}-c\right) +\frac{1}{2}\gamma ^{2}+(2-\gamma )^{2}=0. \end{aligned}$$

We have that $h_{II}(0)<0$, $h_{II}(1)<0$, $h_{II}(\gamma )>0$ for some $\gamma \in (0,1)$, and there exists a threshold value $\widetilde{\gamma } \in (0,1)$ such that $h_{II}^{\prime }(\gamma )\gtreqless 0$ as $\gamma \lesseqgtr \widetilde{\gamma }$. Therefore, there is one root to $h_{II}(\gamma )=0$ on the interval $(0,\widetilde{\gamma })$, and there is another root to $h_{II}(\gamma )=0$ on the interval $(\widetilde{\gamma },1)$ . Hence, here there exist two crossing points at which $\gamma ^{*}=\gamma ^{w}$ when $\theta$ approaches $\theta _{w}$. Denote those crossing points by $\gamma (f_{II}^{a})$ and $\gamma (f_{II}^{b})$, where $f_{II}^{a}<f_{II}^{b}$. By continuity, this shows part (ii.3) and completes the proof.