Ecolabel: Is More Information Better?

Abstract

We study in this paper the effect of the type of information provided by an ecolabel. For this purpose, in the framework of a model of vertical differentiation, we compare the effects of a partial information label (Type I) and a complete information label (Type III) on firms’ profits, industry profit, consumers’ surplus, environmental damage and social welfare. A partial information label indicates that the environmental quality of a good exceeds some given threshold. The authority issuing a partial information label chooses its labeling criteria while maximizing the social welfare. A complete information label indicates the exact environmental quality chosen by firms. We prove that while a partial information label always improves the social welfare and deteriorates the green firm profit compared to a complete information label, the comparison between the two types of ecolabel in terms of the brown firm’s profit, the industry’s profit, the consumers surplus and the environment depends in a non-obvious way on the marginal cost of quality and on the environmental sensitivity to quality.

Appendices

Appendix A

1.1 Proof of Lemma 1

The proof follows from Result 1. The firms’ demands and margins with intermediate marginal costs are given, respectively, by:

${\left\{ \begin{array}{ll} D_1=\frac{\overline{\theta }-2\underline{\theta }+\alpha }{3},\\ D_2=\frac{2\overline{\theta }-\underline{\theta }-\alpha }{3}.\\ \end{array}\right. }$ and ${\left\{ \begin{array}{ll} p_1-\alpha q_1=\frac{\overline{\theta }-2\underline{\theta }+\alpha }{3}(q_2-q_1),\\ p_2-\alpha q_2=\frac{2\overline{\theta }-\underline{\theta }-\alpha }{3}(q_2-q_1).\\ \end{array}\right. }$

$\blacksquare$

1.2 Proof of Result 2

The proof is based on Result 1. Assuming that $q_1<q_2$, we have that $\Pi _1=(\alpha -\overline{\theta })(q_2-q_1)-\beta (q_1-\underline{q})^2$ and $\Pi _2=-\beta (q_2-\underline{q})^2$. We easily check that $\Pi _1$ is decreasing with respect to $q_1$ and $\Pi _2$ is decreasing with respect to $q_2$. Thus, both firms produce $\underline{q}$. When $q_1=q_2=q$ then $\Pi _1=\Pi _2=-\beta (\Delta q)^2$ and both firms produce $\underline{q}$. $\blacksquare$

Denote by $\underline{\delta }_I=\frac{(\alpha +\overline{\theta }-2\underline{\theta })^2}{9\beta }$.

Denote by $\underline{D}$, $\underline{CS}$ and $\underline{SW}$ the environmental damage, consumers’ surplus and the social welfare, respectively, when both firms produce $\underline{q}$. They are given as follows:

$$\underline{D}=(\gamma -\mu \underline{q})(\overline{\theta }-\underline{\theta }),$$

$$\underline{CS}=\int _{\underline{\theta }}^{\overline{\theta }} (\theta \underline{q}-\alpha \underline{q})d\theta =(\overline{\theta }-\underline{\theta })(\frac{\overline{\theta }+\underline{\theta }}{2}-\alpha )\underline{q},$$

$$\underline{SW}=\underline{CS}-(\gamma -\mu \underline{q})(\overline{\theta }-\underline{\theta })=(\overline{\theta }-\underline{\theta })((\frac{\overline{\theta }+\underline{\theta }}{2}-\alpha +\mu )\underline{q}-\gamma ).$$

Denote by $\Pi _1$, $\Pi _2$, $\Pi$, D, CS and SW, respectively, the brown firm’s profit, the green firm’s profit, the industry profit, the environmental damage, consumers’ surplus and the social welfare. Result 3 gives their general expressions. Whether in the partial information label case or in the complete information one, Lemmas 2 and 3 show that either both firms produce $\underline{q}$ or one firm remains brown and the other firm becomes green.

Result 3

The general expression of firms’ profits, Industry profit, environmental damage, consumers’ surplus and the social welfare at price equilibrium when one firm produces $\underline{q}$ and the other firm produces $q\ge \underline{q}$ are given in Table 2.

Table 2 The firms’ profits, the industry profit, the environmental damage, the consumers’ surplus and the social welfare when one firm produces $\underline{q}$ and the other q

1.3 Proof of Result 3

We distinguish three cases:

• 1. When $\alpha \le 2\underline{\theta }-\overline{\theta }$, as proved in Result 1, only the green firm is active. The expressions of firms’ profits, industry profit, environmental damage, consumers’ surplus and the social welfare, for a given q produced by the green firm, are as follows:

  $$\Pi _2=(\underline{\theta }-\alpha )(\overline{\theta }-\underline{\theta })\Delta q-\beta \Delta q^2=\beta (\overline{\delta }_L-\Delta q)\Delta q,$$
  $$\Pi _{1}=0,$$
  $$\Pi =\Pi _1+\Pi _2=\beta (\overline{\delta }_L-\Delta q)\Delta q,$$
  $$D=(\gamma -\mu q)(\overline{\theta }-\underline{\theta })=\underline{D}-\mu \Delta q(\overline{\theta }-\underline{\theta }),$$
  $$\begin{aligned}CS&=\int _{\underline{\theta }}^{\overline{\theta }}(\theta q-\alpha \underline{q}-\underline{\theta }\Delta q)d\theta \\&=\frac{(\overline{\theta }-\underline{\theta })^2}{2}q-(\alpha -\underline{\theta })(\overline{\theta }-\underline{\theta })\underline{q}\\&=\underline{CS}+\frac{(\overline{\theta }-\underline{\theta })^2}{2}\Delta q,\end{aligned}$$
  $$\begin{aligned}SW&=CS+\Pi _1+\Pi _{2}-(\gamma -\mu q)(\overline{\theta }-\underline{\theta })\\ &=(\overline{\theta }-\underline{\theta })[(\frac{\overline{\theta }+\underline{\theta }}{2}-\alpha -\mu ) q-\gamma ]-\beta \Delta q^2, \\& =\underline{SW}+\beta \Delta q(2\delta ^*_L-\Delta q).\end{aligned}$$

• 2. When $2\underline{\theta }-\overline{\theta }<\alpha <2\overline{\theta }-\underline{\theta }$ and one firm is green, i.e. $q>\underline{q}$, the expressions of firms’ profits, industry profit, environmental damage, consumers’ surplus and the social welfare for a given q are as follows:

  $$\Pi _2=\frac{1}{9}(2\overline{\theta }-\underline{\theta }-\alpha )^2\Delta q-\beta \Delta q^2=\beta (\overline{\delta }_I-\Delta q)\Delta q,$$
  $$\Pi _{1}=\frac{1}{9}(\alpha +\overline{\theta }-2\underline{\theta })^2\Delta q=\beta \underline{\delta }_I\Delta q,$$
  $$\Pi =\Pi _1+\Pi _2=\beta (\overline{\delta }_I+\underline{\delta }_I-\Delta q)\Delta q,$$
  $$\begin{aligned}D=&(\gamma -\mu \tilde{q})\frac{2\overline{\theta }-\underline{\theta }-\alpha }{3}+(\gamma -\mu \underline{q})\\ &\frac{\alpha +\overline{\theta }-2\underline{\theta }}{3}=\underline{D}-\mu \Delta q\frac{2\overline{\theta }-\underline{\theta }-\alpha }{3},\end{aligned}$$
  $$\begin{aligned}CS=&\int _{\underline{\theta }}^{\hat{\theta }}(\theta \underline{q}-\frac{1}{3}(2\alpha \underline{q}+\alpha q+(\overline{\theta }-2\underline{\theta })\Delta q))d\theta \\ &+\int _{\hat{\theta }}^{\overline{\theta }}(\theta q-\frac{1}{3}(\alpha \underline{q}+2\alpha q+(2\overline{\theta }-\underline{\theta })\Delta q))d\theta ,\end{aligned}$$
  $$=\underline{CS}+\Delta q(\frac{(\underline{\theta }-\alpha )(2\overline{\theta }-\underline{\theta }-\alpha )}{6}-\frac{(\overline{\theta }-2\underline{\theta }+\alpha )^2}{9}),$$
  $$SW=CS+\Pi _1+\Pi _{2}-(\gamma -\mu q)(\frac{2\overline{\theta }-\underline{\theta }-\alpha }{3})-(\gamma -\mu \underline{q})\frac{\alpha +\overline{\theta }-2\underline{\theta }}{3},$$
  $$=\underline{SW}+\beta \Delta q(2\delta ^*_I-\Delta q).$$

• 3. When $2\underline{\theta }-\overline{\theta }<\alpha <2\overline{\theta }-\underline{\theta }$ and both firms are brown then $\Pi _1=\Pi _2=\Pi =0$; $D=\underline{D}$, $CS=\underline{CS}$ and $SW=\underline{SW}$. The general expression established in the previous case are still valid in this case with $\Delta q=0$. $\blacksquare$

Result 4

In the partial information label case, depending on the marginal cost $\alpha$ and the environmental sensitivity to quality $\mu$, at the SPNE, the firms’ profits, the industry profit, the environmental damage, consumers’ surplus and the social welfare are given in Table 3 for low $\alpha$ and Table 4 for intermediate $\alpha$.

Table 3 The SPNE outcome for low $\alpha$ with a partial information label

Table 4 The SPNE outcome for intermediate $\alpha$ with a partial information label

1.4 Proof of Result 4

We substitute $\Delta q$ in the general expressions given in Table 2 by the values provided in Lemma 2. $\blacksquare$

1.5 Proof of Lemma 3

Using Result 1, we distinguish two cases:

• If  $\alpha \le 2\underline{\theta }-\overline{\theta }$, firms’ profits are given by:

  $$\Pi _1=-\beta (q_1-\underline{q})^2,$$
  $$\Pi _2=(\underline{\theta }-\alpha )(\overline{\theta }-\underline{\theta })(q_2-q_1)-\beta (q_2-\underline{q})^2.$$

  We check that $\frac{\partial \Pi _1}{\partial q_1}=-2\beta (q_1-\underline{q})<0$. Thus, Firm 1 chooses $q^*_1=\underline{q}$. We check that $\Pi _2$ is concave down with respect to $q_2$ and that it is maximized at $q^*_2=\frac{\overline{\delta }_L}{2}+\underline{q}$.

• If  $2\underline{\theta }-\overline{\theta }<\alpha <2\overline{\theta }-\underline{\theta }$, firms’ profits are given by:

  $$\Pi _1=\frac{1}{9}(2\underline{\theta }-\overline{\theta }-\alpha )^2(q_2-q_1)-\beta (q_1-\underline{q})^2,$$
  $$\Pi _2=\frac{1}{9}(2\overline{\theta }-\underline{\theta }-\alpha )^2(q_2-q_1)-\beta (q_2-\underline{q})^2.$$

  We check that $\frac{\partial \Pi _1}{\partial q_1}=-\frac{1}{9}(2\underline{\theta }-\overline{\theta }-\alpha )^2-2\beta (q_1-\underline{q})<0$. Thus, Firm 1 chooses $q^*_1=\underline{q}$. We check that $\Pi _2$ is concave down with respect to $q_2$ and that it is maximized at $q^*_2=\frac{\overline{\delta }_I}{2}+\underline{q}$. $\blacksquare$

Result 5

In the complete information label case, depending on the marginal cost $\alpha$, at the SPNE, the firms’ profits, the industry profit, the environmental damage, consumers’ surplus and the social welfare are given in Table 5.

Table 5 The SPNE outcome with a complete information label

1.6 Proof of Result 5

We substitute $\Delta q$ in the general expressions given by Table 2 by the values provided in Lemma 3. $\blacksquare$

Denote by $SW^p$, $SW^c$, the social welfare with a partial and complete information label, respectively. Denote by $\Pi ^p_2$ and $\Pi ^c_2$, the green firm profit with a partial and complete information label, respectively.

1.7 Proof of Proposition 1

We start by comparing the social welfare in both cases:

• If $\alpha \le 2\underline{\theta }-\overline{\theta }$, then

  • if $\mu <\frac{3\underline{\theta }-\overline{\theta }}{2}-\alpha$, then ${SW}^p-{SW}^c=\beta (\delta ^*_L-\frac{\overline{\delta }_L}{2})^2>0$.

  • if $\mu >\frac{3\underline{\theta }-\overline{\theta }}{2}-\alpha$, then ${SW}^p-{SW}^c=\beta \overline{\delta }_L(\delta ^*_L-\frac{3}{4}\overline{\delta }_L)>0$ as $\delta ^*_L>\overline{\delta }_L$.

• If $2\underline{\theta }-\overline{\theta }<\alpha <2\overline{\theta }-\underline{\theta }$

  • if $\mu <\frac{5\alpha -4\overline{\theta }-\underline{\theta }}{6}$, then ${SW}^p-{SW}^c=-\beta \overline{\delta }_I(\delta ^*_I-\frac{\overline{\delta }_I}{4})>0$ as $\delta ^*_I<0$ when $\mu <\frac{5\alpha -4\overline{\theta }-\underline{\theta }}{6}$.

  • if $\frac{5\alpha -4\overline{\theta }-\underline{\theta }}{6}<\mu <\frac{4\overline{\theta }-5\underline{\theta }+\alpha }{6}$, then ${SW}^p-{SW}^c=\beta (\delta ^*_I-\frac{\overline{\delta }_I}{2})^2>0$.

  • if $\mu >\frac{4\overline{\theta }-5\underline{\theta }+\alpha }{6}$, then ${SW}^p-{SW}^c=\beta \overline{\delta }_I(\delta ^*_I-\frac{3}{4}\overline{\delta }_I)>0$ as $\delta ^*_I>\overline{\delta }_I$.

We now move to the comparison of the green firm profits

• If $\alpha \le 2\underline{\theta }-\overline{\theta }$, then

  • if $\mu <\frac{3\underline{\theta }-\overline{\theta }}{2}-\alpha$, then $\Pi ^p_2-\Pi ^c_2=-\beta (\delta ^*_L-\frac{\overline{\delta }_L}{2})^2<0$.

  • if $\mu >\frac{3\underline{\theta }-\overline{\theta }}{2}-\alpha$, then $\Pi ^p_2-\Pi ^c_2=-\beta \frac{(\overline{\delta }_L)^2}{4}<0$.

• If $2\underline{\theta }-\overline{\theta }<\alpha <2\overline{\theta }-\underline{\theta }$, then

  • if $\mu <\frac{5\alpha -4\overline{\theta }-\underline{\theta }}{6}$, then $\Pi ^p_2-\Pi ^c_2=-\beta \frac{(\overline{\delta }_I)^2}{4}<0$.

  • if $\frac{5\alpha -4\overline{\theta }-\underline{\theta }}{6}<\mu <\frac{4\overline{\theta }-5\underline{\theta }+\alpha }{6}$, then $\Pi ^p_2-\Pi ^c_2=-\beta (\delta ^*_I-\frac{\overline{\delta }_I}{2})^2<0$.

  • if $\mu >\frac{4\overline{\theta }-5\underline{\theta }+\alpha }{6}$, then $\Pi ^p_2-\Pi ^c_2= -\beta \frac{(\overline{\delta }_I)^2}{4}<0$. $\blacksquare$

1.8 Proof of Proposition 2

The cases that appeared when the label is a partial information label have to be distinguished here:

• If $\alpha \le 2\underline{\theta }-\overline{\theta }$, then

  • if $\mu <\frac{3\underline{\theta }-\overline{\theta }}{2}-\alpha$, then we easily prove that $\delta ^*_L>\frac{\overline{\delta }_L}{2}$. Indeed, $\delta ^*_L-\frac{\overline{\delta }_L}{2}=\frac{(\overline{\theta }-\underline{\theta })(\overline{\theta }-\underline{\theta }+2\mu )}{4\beta }>0$.

  • if $\mu >\frac{3\underline{\theta }-\overline{\theta }}{2}-\alpha$, then obviously $\overline{\delta }_L>\frac{\overline{\delta }_L}{2}$.

• If $2\underline{\theta }-\overline{\theta }<\alpha <2\overline{\theta }-\underline{\theta }$, then

  • if $\mu <\frac{5\alpha -4\overline{\theta }-\underline{\theta }}{6}$, then a partial information labeling authority does not offer any label, i.e. $\Delta \tilde{q}=0$. Thus, $\Delta \tilde{q}=0<\frac{\overline{\delta }_I}{2}$.

  • if $\frac{5\alpha -4\overline{\theta }-\underline{\theta }}{6}<\mu <\frac{4\overline{\theta }-5\underline{\theta }+\alpha }{6}$, then $\delta ^*_I>\frac{\overline{\delta }_I}{2}$ if and only if $\mu >\frac{\alpha -\underline{\theta }}{2}$.

  • if $\mu >\frac{4\overline{\theta }-5\underline{\theta }+\alpha }{6}$, then obviously $\overline{\delta }_I>\frac{\overline{\delta }_I}{2}$. $\blacksquare$

As for the environmental damage, we notice from Table 2 that the environmental damage is linearly and negatively related to the green quality. Denote by $D^p$ and $D^c$, the environmental damage with a partial and complete information label, respectively, we have:

• If $\alpha \le 2\underline{\theta }-\overline{\theta }$, then the partial information labeling criteria are more stringent than the green quality of the complete information label and consequently $D^p-D^c<0$.

• If $2\underline{\theta }-\overline{\theta }<\alpha <2\overline{\theta }-\underline{\theta }$, then the partial information labeling criteria are more stringent than the green quality of the complete information label if and only if $\mu >\frac{\alpha -\underline{\theta }}{2}$ in which case, we have $D^p-D^c>0$.$\blacksquare$

Denote by $\Pi ^p_1$ and $\Pi ^c_1$, the brown firm profit with a partial and complete information label, respectively. Recall that $D^p$ and $D^c$ denote the environmental damage with a partial and complete information label, respectively.

1.9 Proof of Proposition 3

The brown firm profit is null when $\alpha <2\underline{\theta }-\overline{\theta }$ as consumers only buy the green product whatever the label’s type. When $2\underline{\theta }-\overline{\theta }<\alpha <2\overline{\theta }-\underline{\theta }$ then the profit of the brown firm is positively and linearly related to $\Delta q$. Thus, the brown firm prefers that the green firm offers the highest possible quality. Using the results of Proposition 2, we have:

• If $\mu >\frac{\alpha -\underline{\theta }}{2}$, then $\Pi ^p_1-\Pi ^c_1>0$.

• If $\mu <\frac{\alpha -\underline{\theta }}{2}$, then $\Pi ^p_1-\Pi ^c_1<0$. $\blacksquare$

1.10 Proof of Proposition 4

Denote by $CS^p$ and $CS^c$, consumers’ surplus with a partial and complete information label, respectively.

• If $\alpha \le 2\underline{\theta }-\overline{\theta }$, then consumers’ surplus is positively and linearly related to $\Delta q$ as shown in Table 2. As the labeling criteria with a partial information label are more stringent than the green quality of the complete information label then ${CS}^p-{CS}^c>0$

• If $2\underline{\theta }-\overline{\theta }<\alpha <2\overline{\theta }-\underline{\theta }$, then consumers’ surplus is linearly related to $\Delta q$ as shown in Table 2. However, the relationship can be positive or negative depending on the sign of $\frac{(\underline{\theta }-\alpha )(2\overline{\theta }-\underline{\theta }-\alpha )}{6}-\frac{(\overline{\theta }-2\underline{\theta }+\alpha )^2}{9}$. We show that if $2\underline{\theta }-\overline{\theta }<\alpha <\tilde{\alpha }$, then $\frac{(\underline{\theta }-\alpha )(2\overline{\theta }-\underline{\theta }-\alpha )}{6}-\frac{(\overline{\theta }-2\underline{\theta }+\alpha )^2}{9}>0$ and if $\tilde{\alpha }<\alpha <2\overline{\theta }-\underline{\theta }$, then$\frac{(\underline{\theta }-\alpha )(2\overline{\theta }-\underline{\theta }-\alpha )}{6}-\frac{(\overline{\theta }-2\underline{\theta }+\alpha )^2}{9}<0$. Thus,

  • If $\mu <\frac{\alpha -\underline{\theta }}{2}$, then $\alpha >\tilde{\alpha }$. Thus, $CS^p-CS^c>0$ as the partial information label is less stringent than the green quality of the complete information label and as consumers’ surplus is negatively related to $\Delta q$.

  • If $\mu >\frac{\alpha -\underline{\theta }}{2}$ and $\alpha <\tilde{\alpha }$, then $CS^p-CS^c>0$ as the partial information label is more stringent than the green quality of the complete information label and as consumers’ surplus is positively related to $\Delta q$.

  • If $\mu >\frac{\alpha -\underline{\theta }}{2}$ and $\alpha >\tilde{\alpha }$, then $CS^p-CS^c<0$ as the partial information label is more stringent than the green quality of the complete information label and as consumers’ surplus is negatively related to $\Delta q$. $\blacksquare$

1.11 Proof of Proposition 5

Denote by $\Pi ^p$ and $\Pi ^c$, the industry profit with a partial and complete information label, respectively.

• If $\alpha \le 2\underline{\theta }-\overline{\theta }$, then the industry profit is equal to the green firm profit and is higher with a complete information label as shown in Proposition 1.

• If $2\underline{\theta }-\overline{\theta }<\alpha <2\overline{\theta }-\underline{\theta }$, then

  • if $\mu <\frac{5\alpha -4\overline{\theta }-\underline{\theta }}{6}$, then $\Pi ^p-\Pi ^c=-\beta (\underline{\delta }_I+\frac{\overline{\delta }_I}{2})\frac{\overline{\delta }_I}{2}<0$.

  • if $\frac{5\alpha -4\overline{\theta }-\underline{\theta }}{6}<\mu <\frac{4\overline{\theta }-5\underline{\theta }+\alpha }{6}$, then $\Pi ^p-\Pi ^c=\beta (\delta ^*_I-\frac{\overline{\delta }_I}{2})(\frac{\overline{\delta }_I}{2}+\underline{\delta }_I-\delta ^*_I)$. We have that $\Pi ^p-\Pi ^c>0$ if and only if $\frac{\overline{\delta }_I}{2}<\delta ^*_I<\frac{\overline{\delta }_I}{2}+\underline{\delta }_I$ which is equivalent to $\frac{\alpha -\underline{\theta }}{2}<\mu <\frac{2}{3}\frac{(\alpha +\overline{\theta }-2\underline{\theta })^2}{2\overline{\theta }-\underline{\theta }-\alpha }+\frac{\alpha -\underline{\theta }}{2}$.

  • if $\mu >\frac{4\overline{\theta }-5\underline{\theta }+\alpha }{6}$, then $\Pi ^p-\Pi ^c=\beta \frac{\overline{\delta }_I}{2}(\underline{\delta }_I-\frac{\overline{\delta }_I}{2})$. We have that $\Pi ^p-\Pi ^c>0$ if and only if $\underline{\delta }_I>\frac{\overline{\delta }_I}{2}$ which is equivalent to $\alpha ^2-2\alpha (5\underline{\theta }-4\overline{\theta })-2\overline{\theta }^2+7\underline{\theta }^2-4\overline{\theta }\underline{\theta }>0$. The quadratic expression is positive if $\alpha >\overline{\alpha }$ and negative if $\alpha <\overline{\alpha }$. Thus,

    • if $\alpha <\overline{\alpha }$, then $\underline{\delta }_I<\frac{\overline{\delta }_I}{2}$ and $\Pi ^p-\Pi ^c<0$.

    • if $\alpha >\overline{\alpha }$, then $\underline{\delta }_I>\frac{\overline{\delta }_I}{2}$ and $\Pi ^p-\Pi ^c>0$. $\blacksquare$

Remark on the difference of consumers’ surplus and difference of industry profits under the two types of labels

Denote by $\Delta CS(\alpha )=CS^p(\alpha )-CS^c(\alpha )$ the difference between consumers’ surpluses with a partial information label and a complete information label at equilibrium and denote by $\Delta \Pi (\alpha )=\Pi ^p(\alpha )-\Pi ^c(\alpha )$ the difference between industry profits with a partial information label and a complete information label at equilibrium. Using Results 4 and 5, the expressions of $\Delta CS(\alpha )$ and $\Delta \Pi (\alpha )$ for $\frac{\overline{\theta }-\underline{\theta }}{2}<\mu <\frac{3\underline{\theta }-\overline{\theta }}{2}$ are, respectively, given by:

$$\Delta CS(\alpha )={\left\{ \begin{array}{ll} \frac{(\overline{\theta }-\underline{\theta })^3}{8\beta }(\overline{\theta }-\underline{\theta }+2\mu ) \quad \hbox { if } \quad \alpha<\frac{3\underline{\theta }-\overline{\theta }}{2}-\mu ,\\ \frac{(\overline{\theta }-\underline{\theta })^3}{4\beta }(\underline{\theta }-\alpha ) \quad \hbox { if } \quad \frac{3\underline{\theta }-\overline{\theta }}{2}-\mu<\alpha<2\underline{\theta }-\overline{\theta },\\ \frac{(2\overline{\theta }-\underline{\theta }-\alpha )^2}{18\beta }(\frac{(\overline{\theta }-\alpha )(2\overline{\theta }-\underline{\theta }-\alpha )}{6}-\frac{(\overline{\theta }-2\underline{\theta }+\alpha )^2}{9}) \quad \hbox { if } \quad 2\underline{\theta }-\overline{\theta }<\alpha<6\mu -4\overline{\theta }+5\underline{\theta },\\ (\frac{(2\overline{\theta }-\underline{\theta }-\alpha )(4\overline{\theta }+\underline{\theta }-5\alpha +6\mu ))}{36\beta }-\frac{(2\overline{\theta }-\underline{\theta }-\alpha )^2}{18\beta })(\frac{(\overline{\theta }-\alpha )(2\overline{\theta }-\underline{\theta }-\alpha )}{6}-\frac{(\overline{\theta }-2\underline{\theta }+\alpha )^2}{9}) \quad \hbox { if } \quad 6\mu -4\overline{\theta }+5\underline{\theta }<\alpha <\frac{6\mu +4\overline{\theta }+\underline{\theta }}{5} ,\\ -\frac{(2\overline{\theta }-\underline{\theta }-\alpha )^2}{18\beta }(\frac{(\underline{\theta }-\alpha )(2\overline{\theta }-\underline{\theta }-\alpha )}{6}-\frac{(\overline{\theta }-2\underline{\theta }+\alpha )^2}{9}) \quad \hbox { if } \quad \alpha >\frac{6\mu +4\overline{\theta }+\underline{\theta }}{5}. \end{array}\right. }$$

$$\Delta \Pi (\alpha )={\left\{ \begin{array}{ll} -\frac{(\overline{\theta }-\underline{\theta })^2}{16\beta }(\overline{\theta }-\underline{\theta }+2\mu )^2 \quad \hbox { if } \quad \alpha<\frac{3\underline{\theta }-\overline{\theta }}{2}-\mu ,\\ -\frac{(\overline{\theta }-\underline{\theta })^2}{4\beta }(\underline{\theta }-\alpha )^2 \quad \hbox { if } \quad \frac{3\underline{\theta }-\overline{\theta }}{2}-\mu<\alpha<2\underline{\theta }-\overline{\theta },\\ \frac{(2\overline{\theta }-\underline{\theta }-\alpha )^2}{18\beta }(\frac{(\alpha +\overline{\theta }-2\underline{\theta })^2}{9}-\frac{(2\overline{\theta }-\underline{\theta }-\alpha )^2}{18}) \quad \hbox { if } \quad 2\underline{\theta }-\overline{\theta }<\alpha<6\mu -4\overline{\theta }+5\underline{\theta },\\ (\frac{(2\overline{\theta }-\underline{\theta }-\alpha )(4\overline{\theta }+\underline{\theta }-5\alpha +6\mu ))}{36}-\frac{(2\overline{\theta }-\underline{\theta }-\alpha )^2}{18})(-\frac{(2\overline{\theta }-\underline{\theta }-\alpha )(4\overline{\theta }+\underline{\theta }-5\alpha +6\mu )}{36\beta }+\frac{(\overline{\theta }-2\underline{\theta }+\alpha )^2}{9\beta }+\frac{(2\overline{\theta }-\underline{\theta }-\alpha )^2}{18\beta }) \quad \hbox { if } \quad 6\mu -4\overline{\theta }+5\underline{\theta }<\alpha <\frac{6\mu +4\overline{\theta }+\underline{\theta }}{5} ,\\ -\frac{(2\overline{\theta }-\underline{\theta }-\alpha )^2}{18\beta }(\frac{(\alpha +\overline{\theta }-2\underline{\theta })^2}{9}+\frac{(2\overline{\theta }-\underline{\theta }-\alpha )^2}{18}) \quad \hbox { if } \quad \alpha >\frac{6\mu +4\overline{\theta }+\underline{\theta }}{5}. \end{array}\right. }$$

Figure 2 gives the curves of $\Delta CS(\alpha )$ and $\Delta \Pi (\alpha )$ in the particular case where $\overline{\theta }=2.5$, $\underline{\theta }=1.5$, $\beta =1$ and $\mu =0.75$.

Fig. 2

Difference of consumers’ surplus and difference of industry profit curves for $\overline{\theta }=2.5$, $\underline{\theta }=1.5$, $\beta =1$ and $\mu =0.75$

Appendix B

Result 6

Under uncovered market, at the SPNE of the complete information label, qualities, prices, profits, Consumers’ Surplus and Social Welfare are given as follows:

• Qualities: $\hat{q}_1^C = \frac{(\overline{\theta } - \alpha )^2 k^2 (4k - 7)}{(4k -1)^3}$, $\hat{q}_2^C = \frac{(\overline{\theta } - \alpha )^2 k^3 (4k - 7)}{(4k -1)^3}.$

• Prices: $\hat{p}_1^C = \frac{(\overline{\theta } - \alpha )^2 k^2 (4k - 7) (\overline{\theta } (k-1) + 3 \alpha k)}{(4k -1)^4}$, $\hat{p}_2^C = \frac{(\overline{\theta } - \alpha )^2 k^3 (4k - 7)(2 \overline{\theta } (k-1) + \alpha (2k +1))}{(4k -1)^4}$

• Profits: $\hat{\pi }_1^C = \frac{(\overline{\theta } - \alpha )^4 k^3 (4k - 7)(4 k^2 -3 k +2)}{2(4k -1)^6}$, $\hat{\pi }_2^C = \frac{8 (\overline{\theta } - \alpha )^4 k^3 (4k - 7)(4 k^2 -3 k +2)}{(4k -1)^6}.$

• Consumers’ Surplus: $\widehat{CS}^C = \frac{(\overline{\theta } - \alpha )^4 k^4 (4k - 7)(4 k +5)}{2(4k -1)^5}$.

• Social Welfare: $\widehat{SW}^C = \frac{(\overline{\theta } - \alpha )^4 k^3 (4k - 7)}{(4k -1)^5} (k(2k +1) + \frac{17 (4k^2 -3 k +2}{2(4k-1)})$.

1.1 Proof of Result 6

Actually the calculation of the quality choice amounts to the solution obtained by Liao [6] replacing $\overline{\theta }$ by $\overline{\theta } - \alpha$.

We solve the game by backward induction, beginning with the price step.

Profits are given as follows:

$${\left\{ \begin{array}{ll} \pi _1 = (p_1 - \alpha q_1)(\frac{p_2 - p_1}{q_2 - q_1} - \frac{p_1}{q_1}) \\ \pi _2 = (p_2 - \alpha q_2)(\overline{\theta } - \frac{p_2 - p_1}{q_2 - q_1}). \end{array}\right. }$$

First-Order Conditions yield the equilibrium prices:

$$\begin{aligned} {\left\{ \begin{array}{ll} p_1 = \alpha q_1 + \frac{(\overline{\theta } - \alpha ) q_1 (q_2 - q_1)}{4 q_2 - q_1} \\ p_2 = \alpha q_2 + \frac{2 (\overline{\theta } - \alpha ) q_2 (q_2 - q_1)}{4 q_2 - q_1} \end{array}\right. } \end{aligned}$$

(1)

At price equilibrium, the profits, as function of qualities, are given as follows:

$${\left\{ \begin{array}{ll} \pi _1 = \frac{(\overline{\theta } - \alpha )^2 q_1 q_2 (q_2 - q_1)}{(4 q_2 - q_1)^2} - (1/2)q_1^2 \\ \pi _2 = \frac{4(\overline{\theta } - \alpha )^2 q_2^2 (q_2 - q_1)}{(4 q_2 - q_1)^2} - (1/2) q_2 ^2 \end{array}\right. }$$

The expressions above are exactly the expressions of the profits obtained by Liao [6] replacing $\overline{\theta }$ by $\overline{\theta } - \alpha$. Thus the equilibrium qualities are given by:

$${\left\{ \begin{array}{ll} q_1 = \frac{(\overline{\theta } - \alpha )^2 k^2 (4k - 7)}{(4k - 1)^3} \\ q_2= \frac{(\overline{\theta } - \alpha )^2 k^3(4k - 7)}{(4k - 1)^3} \end{array}\right. }$$

Replacing qualities by their values in the equilibrium prices, then in the profits, we obtain the expressions in the result.

As for the consumers’ surplus, we have to do more effort before reaching the result.

Denote by $\theta _{0,1} (\overline{\theta }, \alpha )$ and $\theta _{1,2} (\overline{\theta }, \alpha )$ the marginal consumers, indifferent, respectively, between no purchase and buying product 1, and between product 1 and 2, at the SPNE, as function of the highest type $\overline{\theta }$ and the cost parameter $\alpha$. The solution obtained by Liao [6] corresponds to $\alpha = 0$. Simple calculations yield:

$$\begin{aligned} {\left\{ \begin{array}{ll} \theta _{0,1} (\overline{\theta }, \alpha ) = \alpha + \theta _{0,1} (\overline{\theta } - \alpha , 0) \\ \theta _{1,2} (\overline{\theta }, \alpha ) = \alpha + \theta _{1,2} (\overline{\theta } - \alpha , 0) \end{array}\right. } \end{aligned}$$

(2)

The consumers’ surplus is given by:

$$CS = \int _{\theta _{0,1} (\overline{\theta }, \alpha )}^{\theta _{1,2} (\overline{\theta }, \alpha )} (\theta q_1 - p_1) d\theta + \int _{\theta _{1,2} (\overline{\theta }, \alpha )}^{\overline{\theta }} (\theta q_2 - p_2) d\theta ,$$

which may be re-written as ($D_i$ standing for the firm i’s demand):

$$\begin{aligned}CS = &(q_1/2) \left( \left( \theta _{1,2} (\overline{\theta }, \alpha ) \right) ^2 - \left( \theta _{0,1} (\overline{\theta }, \alpha ) \right) ^2\right) - p_1 D_1 \\ &+ (q_2/2) \left( \overline{\theta }^2 - \left( \theta _{1,2} (\overline{\theta }, \alpha )\right) ^2\right) - p_2 D_2.\end{aligned}$$

Rearranging the first and third terms using the formula $A^2 - B^2 = (A-B)(A+B)$, then using the relations between the marginal consumers given by equations (2), we obtain:

$$\begin{aligned} CS = (q_1/2) \left( \left( \theta _{1,2} (\overline{\theta } - \alpha , 0) \right) ^2 - \left( \theta _{0,1} (\overline{\theta } - \alpha , 0) \right) ^2 \right) - (p_1 - \alpha q_1) D_1 \\ + (q_2/2) \left( \overline{\theta }^2 - \left( \theta _{1,2} (\overline{\theta } - \alpha , 0)\right) ^2 \right) - (p_2 - \alpha q_2) D_2. \end{aligned}$$

The second and fourth terms correspond, respectively, to $- \pi _1 (\overline{\theta }, \alpha )$ and $- \pi _2 (\overline{\theta }, \alpha )$. But the profits at the SPNE with any $\alpha$ may be related to the profits with $\alpha = 0$ obtained by Liao [6], as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} \pi _1 (\overline{\theta }, \alpha ) = \pi _1 (\overline{\theta } - \alpha , 0) \\ \pi _2 (\overline{\theta }, \alpha ) = \pi _2 (\overline{\theta } - \alpha , 0) \end{array}\right. } \end{aligned}$$

(3)

Hence, the consumers’ surplus with any $\alpha$ can be related to the consumers’ surplus with $\alpha = 0$ obtained by Liao [6] as follows:

$CS (\overline{\theta }, \alpha ) = CS (\overline{\theta } - \alpha , 0),$ which gives the expression provided in the result.

As for SW, we have $SW (\overline{\theta }, \alpha ) = CS (\overline{\theta }, \alpha ) + \pi _1 (\overline{\theta }, \alpha ) + \pi _2 (\overline{\theta }, \alpha ).$ Using the results which have just been proved on CS and the profits, we have: $SW(\overline{\theta }, \alpha ) = SW(\overline{\theta }- \alpha , 0),$ thus the expression.

$\blacksquare$

Result 7

Under the partial information label with uncovered market, the labeling criteria, the qualities, the prices, the profits, the Consumers’ Surplus and the Social Welfare are given as follows:

• Labeling criteria: $\hat{q} = \frac{3 (\overline{\theta }- \alpha )^2}{8}.$

• Qualities: $\hat{q}_2^P = \hat{q}$ and $\hat{q}_1^P = 0$.

• Prices: $\hat{p}_1^P = 0$, $\hat{p}_2^P= (3/16) (\overline{\theta }- \alpha )^2 (\overline{\theta } + \alpha )$.

• Profits: $\hat{\pi }_1^P = 0$, $\hat{\pi }_2^P = (3/128) (\overline{\theta }- \alpha )^4$.

• Consumers’ Surplus: $\widehat{CS}^P= (3/64) (\overline{\theta }- \alpha )^4$.

• Social Welfare: $\widehat{SW}^P = (9/128) (\overline{\theta }- \alpha )^4$.

1.2 Proof of Result 7

We first determine the equilibrium qualities for a given level of labeling criteria.

Using the equilibrium prices obtained in Equations (1), we build a payment matrix with $\{0, \tilde{q} \}$ as the strategy space of each firm. Simple calculations yield the equilibrium qualities for given $\tilde{q}$:

• If  $\tilde{q} > \frac{(\overline{\theta } - \alpha )^2}{2}$, then the quality equilibrium is given by (0, 0).

• If  $\tilde{q} \le \frac{(\overline{\theta } - \alpha )^2}{2}$, then there are two quality equilibria: $(0, \tilde{q})$ and $(\tilde{q}, 0)$.

The labeling criteria must maximize the Social Welfare at quality equilibrium:

$$SW = {\left\{ \begin{array}{ll} (3/8) (\overline{\theta }- \alpha )^2 \tilde{q} - (1/2) \tilde{q}^2 &{} \text{ if } \tilde{q} \le \frac{(\overline{\theta } - \alpha )^2}{2}\\ 0 &{} \text{ otherwise } \end{array}\right. }$$

The maximization of SW above yields the value of $\tilde{q}= \frac{3 (\overline{\theta }- \alpha )^2}{8}$. Replacing $\tilde{q}$ by the obtained value in the equilibrium qualities, then in the prices, the profits, the Consumers’ Surplus and the Social Welfare, we obtain the expressions given in the result. $\blacksquare$

1.3 Proof of Proposition 6

The proof follows from straightforward calculations using Results 6 and 7. $\blacksquare$

1.4 Proof of Proposition 7

The social welfare at equilibrium under the partial information label and covered market $SW^P$ is obtained from Table 4 in Result 4, substituting $\underline{\theta }$, $\mu$ and $\underline{q}$ by 0 and $\beta$ by $\frac{1}{2}$, as follows:

$$SW^P={\left\{ \begin{array}{ll} 0 &{} \text{ if } \alpha >\frac{4}{5}\overline{\theta },\\ \frac{1}{2}(\frac{(2\overline{\theta }-\alpha )(4\overline{\theta }-5\alpha )}{18})^2 &{} \text{ if } \alpha <\frac{4}{5}\overline{\theta }.\\ \end{array}\right. }$$

Simple comparisons with the expression of $\widehat{SW}^P$ at equilibrium under uncovered market provided in Result 7 lead to the proposition conclusions. $\blacksquare$