The value of personal information in vertically differentiated markets with privacy concerns

Abstract

Information technology and data science have enabled firms to practice price discrimination on an unprecedented scale, arousing privacy concerns among their customers. When consumers know a firm is practicing price discrimination, they may take costly measures to conceal their identities so as to avoid being targeted. Governments, in turn, may require firms to disclose their price discrimination practices in order to protect consumers’ interests. In this paper, we consider a pricing game in which two competitive, vertically differentiated firms may implement price discrimination using information purchased from a third-party data supplier. We determine (1) the firms’ optimal pricing strategies when consumers can (or cannot) safeguard their personal information by paying a “privacy cost”; (2) the data supplier’s optimal sales strategy and the value of the data; and (3) the effects of the cost of consumer privacy and of the disclosure of price discrimination practices on firms and consumers. We find that for the data supplier, the optimal sales strategy is always to sell exclusively to one firm, regardless of whether consumers are aware that the firm practices “personalized pricing”. The question of which firm the data broker should sell to depends on what we term the “quality-adjusted cost”—the ratio between the additional cost of the high-quality product and the magnitude of the quality difference. If this ratio is smaller than 1/2, the data broker will sell to the high-quality firm; if greater, to the low-quality firm. Second, by comparing two scenarios involving the disclosure or non-disclosure of price discrimination, we find, somewhat counter-intuitively, that mandatory transparency increases industry profits and decreases consumer surplus when only the high-quality firm has access to consumer data. When only the low-quality firm has such access, transparency lowers industry profits once the quality-adjusted cost exceeds a certain threshold. When the quality-adjusted cost is in the intermediate range, mandatory transparency decreases social welfare. This means that the disclosure of price discrimination practices may have unfavorable consequences from a social planning standpoint. Thus, the new insights our findings offer into competitive personalized pricing in vertically differentiated markets will be useful not only to managers in the industry but also to regulators.

Appendix

Proof of Proposition 1

When both firms choose to buy information, the tailored price is similar to that found in previous studies (Choudary 2005; Tayler 2014): $p_{L} \left( \theta \right) = c - \theta \Delta \;{\text{and}}\;p_{H} \left( \theta \right) = \theta \Delta$. The equilibrium price is given by the following two functions:

$$ \begin{aligned} & \mathop {{\text{argmax}}}\limits_{{p_{H} }} \left[ {[\left( {1 - \theta _{N} } \right)\left( {p_{H} - c} \right) + \int\limits_{\mu }^{1} {\left( {p_{H} \left( \theta \right) - c} \right)} {\text{d}}\theta } \right]. \\ & \mathop {{\text{argmax}}}\limits_{{p_{L} }} \left[ {\theta _{N} p_{L} + \int\limits_{0}^{\mu } {p_{L} } \left( \theta \right){\text{d}}\theta } \right]. \\ \end{aligned} $$

It is easily proved that both $\pi_{H}$ and $\pi_{L}$ are concave functions. By the first-order derivatives, we have $\frac{{\partial \pi_{H} }}{{p_{H} }} = \frac{{c - 2p_{H} + p_{L} + c}}{\Delta } = 0,\quad , \frac{{\partial \pi_{L} }}{{p_{L} }} = \frac{{p_{H} - 2p_{L} }}{\Delta } = 0$. Solving the two equations, we $p_{H} = \frac{{2\left( {c + \Delta } \right)}}{3}$ and $p_{L} = \frac{c + \Delta }{3}$. The firms’ equilibrium profits are $\pi_{H} = \frac{{11c^{2} - 26c\Delta + 17\Delta^{2} }}{18\Delta } and \pi_{L} = \frac{{11c^{2} + 4{\text{c}}\Delta + 2\Delta^{2} }}{18\Delta }$, respectively. Finally, the total consumer surplus is given by

$$ \begin{aligned} & CS = \mathop \int \limits_{0}^{{\theta_{N} }} \left( {V + \theta q_{L} - p_{L} } \right){\text{d}}\theta + \mathop \int \limits_{{\theta_{N} }}^{1} \left( {{\text{V}} + \theta q_{H} - p_{H} } \right){\text{d}}\theta + \mathop \int \limits_{0}^{\mu } \left( {{\text{V}} + \theta q_{L} - p_{L} \left( \theta \right)} \right){\text{d}}\theta + \mathop \int \limits_{\mu }^{1} \left( {{\text{V}} + \theta q_{H} - p_{H} \left( \theta \right)} \right){\text{d}}\theta \\ & \quad = 2{\text{V}} - \frac{{4c^{2} }}{{9(q_{H} - q_{L)} }} + \frac{{ - q_{H} + 10q_{L} - 5c}}{9}. \\ \end{aligned} $$

Proof of Corollary 1

Compare the profits between the NN case and the BB case in equilibrium. For firm H, we have

$$ \begin{aligned} \pi_{H}^{BB} - \pi_{H}^{NN} &= \frac{{11c^{2} - 26c\Delta + 17\Delta^{2} }}{18\Delta } - \frac{{2\left( {c - 2\Delta {\text{q}}} \right)^{2} }}{9\Delta } \\ & = \frac{{7c^{2} - 10c\Delta + \Delta^{2} }}{18\Delta }\\ & = \frac{\Delta }{18}\left( {7\mu^{2} - 10\mu + 1} \right). \\ \end{aligned} $$

We solve the inequality $7\mu^{2} - 10\mu + 1 \ge 0$ which is due to $\mu \in \left[ {0,1} \right]$. There exists a threshold $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mu } = \frac{{1}}{7}\left( {5 - 3\sqrt 2 } \right) \approx 0.11$. When $\mu < \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mu }$, $\pi_{H}^{BB} - \pi_{H}^{NN} > 0$ and when $\mu \ge \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mu }$, $\pi_{H}^{BB} - \pi_{H}^{NN} \le 0$.

For firm L,

$$ \begin{aligned} \pi_{L}^{BB} - \pi_{L}^{NN} &= \frac{{11c^{2} + 4{\text{c}}\Delta + 2\Delta^{2} }}{18\Delta } - \frac{{2\left( {c + \Delta } \right)^{2} }}{9\Delta } \\ & = \frac{{7c^{2} - 4c\Delta - 2\Delta^{2} }}{18\Delta } \\ & = \frac{\Delta }{18}\left( {7\mu^{2} - 4\mu - 2} \right). \\ \end{aligned} $$

We also solve the inequality $7\mu^{2} - 4\mu - 2 \ge 0$. There exists a threshold $\bar{\mu } = \frac{{1}}{7}\left( {2 + 3\sqrt 2 } \right) \approx 0.89$. When $\mu \ge \bar{\mu }$, $\pi_{L}^{BB} \ge \pi_{L}^{NN}$ and when $\mu < \bar{\mu }, \pi_{L}^{BB} < \pi_{L}^{NN}$. We now obtain the announced result.

Proof of Proposition 2

When only firm H has information, firm H has the exclusive power to price-discriminate and set a tailored price $p_{H} \left( \theta \right) = p_{L} + \theta \Delta$. Correspondingly, firm L has to decide whether to set a new price so as to retain its market share.

Case 1. $\theta_{1} \left( {p_{L} } \right) \ge 0$. Firm L sets a relatively low price in order to defend its market share. To maximize the two firms’ profits, we have the following functions:

$$ \begin{aligned} & \mathop {\text{argmax}}\limits_{{p_{H} }} [\left( {1 - \theta_{N} } \right)\left( {p_{H} - c} \right) + \mathop \int \limits_{{\theta_{1} }}^{1} \left( {p_{H} \left( \theta \right) - c} \right){\text{d}}\theta ]. \\ & \mathop {\text{argmax}}\limits_{{p_{L} }} [\theta_{N} p_{L} + \mathop \int \limits_{0}^{{\theta_{1} }} p_{L} {\text{d}}\theta ]. \\ \end{aligned} $$

We can easily prove that both $\pi_{H}$ and $\pi_{L}$ are concave functions. By the first-order derivatives, we have $\frac{{\partial \pi_{H} }}{{p_{H} }} = \frac{{c - 2p_{H} + p_{L} + c}}{\Delta } = 0,\quad \frac{{\partial \pi_{L} }}{{p_{L} }} = \frac{{p_{H} - 4p_{L} + c}}{\Delta } = 0$. Solving the two equations simultaneously, we obtain $p_{H} = \frac{5c + 4\Delta }{7} {\text{and }}p_{L} = \frac{3c + \Delta }{7}$. To ensure the necessary condition $\theta_{1} \left( {p_{L} } \right) = \frac{{c - p_{L} }}{\Delta } \ge 0$, we have ${\text{c}} - \frac{3c + \Delta }{7} \ge 0$ and $\mu = \frac{{c}}{\Delta } \ge \frac{{1}}{4}$. The firms’ profits are, respectively, $\pi_{H} = \frac{{12\left( {c - 2\Delta } \right)^{2} }}{49\Delta }\;{\text{and}}\;\pi_{L} = \frac{{2\left( {3c + \Delta } \right)^{2} }}{49\Delta }$. And the total consumer surplus is given by

$$ \begin{aligned} & CS = \mathop \int \limits_{0}^{{\theta_{N} }} \left( {V + \theta q_{L} - p_{L} } \right){\text{d}}\theta + \mathop \int \limits_{{\theta_{N} }}^{1} \left( {{\text{V}} + \theta q_{H} - p_{H} } \right){\text{d}}\theta + \mathop \int \limits_{0}^{{\theta_{1} }} \left( {{\text{V}} + \theta q_{L} - p_{L} } \right){\text{d}}\theta + \mathop \int \limits_{{\theta_{1} }}^{1} \left( {{\text{V}} + \theta q_{H} - p_{H} \left( \theta \right)} \right){\text{d}}\theta \\ & \quad = 2V + \frac{{4c^{2} }}{{49(q_{H} - q_{L} )}} + \frac{{ - 6q_{H} + 55q_{L} - 50c}}{49}. \\ \end{aligned} $$

Case 2. $\theta_{1} \left( {p_{L} } \right) < 0$. Firm L chooses not to defend its market share. To maximize the two firms’ profits, we have the following functions:

$$ \begin{aligned} & \mathop {{\text{argmax}}}\limits_{{p_{H} }} \left[ {\left( {1 - \theta _{N} } \right)\left( {p_{H} - c} \right) + \int\limits_{0}^{1} {\left( {p_{H} \left( \theta \right) - c} \right)} {\text{d}}\theta } \right]. \\ & \mathop {{\text{argmax}}}\limits_{{p_{L} }} \theta _{N} p_{L} . \\ \end{aligned} $$

We can easily prove that both $\pi_{H}$ and $\pi_{L}$ are concave functions. By the first-order derivatives, we have $\frac{{\partial \pi_{H} }}{{p_{H} }} = \frac{{c - 2p_{H} + p_{L} + c}}{\Delta } = 0,\quad \frac{{\partial \pi_{L} }}{{p_{L} }} = \frac{{p_{H} - 2p_{L} }}{\Delta } = 0$. Solving the two equations simultaneously, we have $p_{H} = \frac{{2\left( {c + \Delta } \right)}}{3} {\text{and }}p_{L} = \frac{c + \Delta }{3}$. To ensure the necessary condition $\theta_{1} \left( {p_{L} } \right) = \frac{{c - p_{L} }}{\Delta } < 0$, we have ${\text{c}} - \frac{c + \Delta }{3} < 0$ and $\mu = \frac{{c}}{\Delta } < \frac{{1}}{2}$. The firms’ profits are expressed by $\pi_{H} = \frac{{2c^{2} - 20c\Delta + 23\Delta^{2} }}{18\Delta }\;and\;\pi_{L} = \frac{{\left( {c + \Delta } \right)^{2} }}{9\Delta }$, respectively And the total consumer surplus is given by

$$ \begin{aligned} & CS = \mathop \int \limits_{0}^{{\theta_{N} }} \left( {V + \theta q_{L} - p_{L} } \right){\text{d}}\theta + \mathop \int \limits_{{\theta_{N} }}^{1} \left( {{\text{V}} + \theta q_{H} - p_{H} } \right){\text{d}}\theta + \mathop \int \limits_{0}^{1} \left( {{\text{V}} + \theta q_{H} - p_{H} \left( \theta \right)} \right){\text{d}}\theta \\ & \quad = 2V + \frac{{c^{2} }}{{18(q_{H} - q_{L} )}} + \frac{{ - 8q_{H} + 26q_{L} - 16c}}{18}. \\ \end{aligned} $$

From the discussion above, it follows that when $\mu < \frac{{1}}{4}$, only case 2 is available, and when $\mu \ge \frac{{1}}{2}$, only case 1 is available. When $\frac{{1}}{4} \le \mu < \frac{{1}}{2}$, firm L may choose between the two cases. Let us compare them to decide which is preferable. We have

$$ \begin{aligned} & \pi_{L}^{case1} - \pi_{L}^{case2} = \frac{{2\left( {3c + \Delta } \right)^{2} }}{49\Delta } - \frac{{\left( {c + \Delta } \right)^{2} }}{9\Delta }. \\ & \quad = \frac{{113c^{2} + 10c\Delta - 31\Delta^{2} }}{441\Delta }. \\ & \quad = \frac{\Delta }{441}\left( {113\mu^{2} + 10\mu - 31} \right). \\ \end{aligned} $$

We solve the inequality $113\mu^{2} + 10\mu - 31 \ge 0$ which is due to $\mu \in \left[ {0,1} \right]$. There exists a threshold $\mu_{1} = \frac{{1}}{113}\left( { - 5 + 42\sqrt 2 } \right) \approx 0.48$. When $\mu \ge \mu_{1}$, $\pi_{L}^{case1} \ge \pi_{L}^{case2}$ and when $\mu < \mu_{1} , \pi_{L}^{case1} < \pi_{L}^{case2}$. Ultimately, the preceding conditions can be expressed as follows: when $0 \le \mu < \mu_{1}$, the equilibrium operates as case 2; when $\mu_{1} \le \mu \le 1$, the equilibrium operates as case 1. Thus we obtain the announced result.

Proof of Proposition 3

When only firm L has information, firm L have the exclusive power to price-discriminate and set a tailored price $p_{L} \left( \theta \right) = p_{H} - \theta \Delta$. Correspondingly, firm H has to decide whether to set a new price so as to retain its market share.

Case 1. $\theta_{2} \left( {p_{H} } \right) \le 1$. Firm H sets a relatively low price to defend its market share. In order to maximize the two firms’ profits, we have the following functions:

$$ \begin{aligned} & \mathop {\text{argmax}}\limits_{{p_{H} }} \left[ {\left( {1 - \theta_{N} } \right)\left( {p_{H} - c} \right) + \left( {1 - \theta_{2} } \right)\left( {p_{H} - c} \right)} \right]. \\ & \mathop {\text{argmax}}\limits_{{p_{L} }} [\theta_{N} p_{L} + \mathop \int \limits_{0}^{{\theta_{2} }} p_{L} \left( \theta \right){\text{d}}\theta ]. \\ \end{aligned} $$

We can easily prove that both $\pi_{H}$ and $\pi_{L}$ are concave functions. By the first-order derivatives, we have $\frac{{\partial \pi_{H} }}{{p_{H} }} = \frac{{2\Delta - 4p_{H} + p_{L} + 2c}}{\Delta } = 0,\quad \frac{{\partial \pi_{L} }}{{p_{L} }} = \frac{{p_{H} - 2p_{L} }}{\Delta } = 0$. Solving the two equations simultaneously, we have $p_{H} = \frac{{4\left( {c + \Delta } \right)}}{7} {\text{and }}p_{L} = \frac{{2\left( {c + \Delta } \right)}}{7}$. To ensure the necessary condition $\theta_{2} \left( {p_{H} } \right) = \frac{{p_{H} }}{\Delta } \le 1$, we have $\frac{{4\left( {c + \Delta } \right)}}{7} - \Delta \le 0$ and then $\mu = \frac{{c}}{\Delta } \le \frac{3}{4}$. The firms’ profits are expressed by $\pi_{H} = \frac{{2\left( {3c - 4\Delta } \right)^{2} }}{49\Delta } and \pi_{L} = \frac{{12\left( {c + \Delta } \right)^{2} }}{49\Delta }$, respectively. And the total consumer surplus is given by

$$ \begin{aligned} & CS = \mathop \int \limits_{0}^{{\theta_{N} }} \left( {V + \theta q_{L} - p_{L} } \right){\text{d}}\theta + \mathop \int \limits_{{\theta_{N} }}^{1} \left( {{\text{V}} + \theta q_{H} - p_{H} } \right){\text{d}}\theta + \mathop \int \limits_{0}^{{\theta_{2} }} \left( {{\text{V}} + \theta q_{L} - p_{L} \left( \theta \right)} \right){\text{d}}\theta + \mathop \int \limits_{{\theta_{2} }}^{1} \left( {{\text{V}} + \theta q_{H} - p_{H} } \right){\text{d}}\theta \\ & \quad = 2V + \frac{{2c^{2} }}{{49\left( {q_{H} - q_{L} } \right)}} + \frac{{ - 5q_{H} + 54q_{L} - 52c}}{49}. \\ \end{aligned} $$

Case 2. $\theta_{2} \left( {p_{H} } \right) > 1$. Firm H chooses not to defend its market share. In order to maximize the two firms’ profits, we have the following functions:

$$ \begin{aligned} & \mathop {\text{argmax}}\limits_{{p_{H} }} \left( {1 - \theta_{N} } \right)\left( {p_{H} - c} \right). \\ & \mathop {\text{argmax}}\limits_{{p_{L} }} [\theta_{N} p_{L} + \mathop \int \limits_{0}^{1} p_{L} \left( \theta \right){\text{d}}\theta ]. \\ \end{aligned} $$

We can easily prove that both $\pi_{H}$ and $\pi_{L}$ are concave functions. By the first-order derivatives, we have $\frac{{\partial \pi_{H} }}{{p_{H} }} = \frac{{c - 2p_{H} + p_{L} + c}}{\Delta } = 0,\quad \frac{{\partial \pi_{L} }}{{p_{L} }} = \frac{{p_{H} - 2p_{L} }}{\Delta } = 0$. Solving the two equations simultaneously, we have $p_{H} = \frac{{2\left( {c + \Delta } \right)}}{3} {\text{and }}p_{L} = \frac{c + \Delta }{3}$. To ensure the necessary condition $\theta_{2} \left( {p_{H} } \right) = \frac{{p_{H} }}{\Delta } > 1$, we have $\frac{{2\left( {c + \Delta } \right)}}{3} - \Delta > 0$ and then $\mu = \frac{{c}}{\Delta } > \frac{{1}}{2}$. The firms profits are expressed by $\pi_{H} = \frac{{\left( {c - 2\Delta } \right)^{2} }}{9\Delta } \,and\,\, \pi_{L} = \frac{{4c^{2} + 20{\text{c}}\Delta + 7\Delta^{2} }}{18\Delta }$. And the total consumer surplus is given by

$$ \begin{aligned} & CS = \mathop \int \limits_{0}^{{\theta_{N} }} \left( {V + \theta q_{L} - p_{L} } \right){\text{d}}\theta + \mathop \int \limits_{{\theta_{N} }}^{1} \left( {{\text{V}} + \theta q_{H} - p_{H} } \right){\text{d}}\theta + \mathop \int \limits_{0}^{1} \left( {{\text{V}} + \theta q_{L} - p_{L} \left( \theta \right)} \right){\text{d}}\theta \\ & \quad = 2V + \frac{{c^{2} }}{{18(q_{H} - q_{L} )}} + \frac{{ - 5q_{H} + 23q_{L} - 22c}}{18}. \\ \end{aligned} $$

From the discussion above, we know that when $\mu \le \frac{{1}}{2}$, only case 1 is available and when $\mu > \frac{3}{4}$, only case 2 is available. When $\frac{{1}}{2} \le \mu < \frac{3}{4}$, firm H has a choice. Let us compare the two cases to decide which is preferable. We calculate

$$ \begin{aligned} & \pi_{H}^{case1} - \pi_{H}^{case2} = \frac{{2\left( {3c - 4\Delta } \right)^{2} }}{49\Delta } - \frac{{\left( {c - 2\Delta } \right)^{2} }}{9\Delta } \\ & \quad = \frac{{113c^{2} - 236c\Delta + 92\Delta^{2} }}{441\Delta } \\ & \quad = \frac{\Delta }{441}\left( {113\mu^{2} - 236\mu + 92} \right). \\ \end{aligned} $$

We solve the inequality $113\mu^{2} - 236\mu + 92 \ge 0$ which is due to $\mu \in \left[ {0,1} \right]$. There exists a threshold $\mu_{2} = \frac{{2}}{113}\left( {59 - 21\sqrt 2 } \right) \approx 0.52$. When $\mu \ge \mu_{2}$, $\pi_{L}^{case1} \le \pi_{L}^{case2}$ and when $\mu \left\langle {\mu_{2} , \pi_{L}^{case1} } \right\rangle \pi_{L}^{case2}$. In summary, the preceding conditions can be expressed as follows: when $\mu \in \left[ {0,\mu_{2} } \right)$, the equilibrium operates as case 1; when $\mu \in \left[ {\mu_{2} ,1} \right]$, the equilibrium operates as case 2.

Proof of Proposition 4

In the preceding discussion, we determined the profits of firm H and firm L in the different cases, as shown in the following chart.

$\mu$	$0 \le \mu < \mu_{1}$	$\mu_{1} \le \mu < \mu_{2}$	$\mu_{2} \le \mu \le 1$
$\pi_{H}^{BN}$	$\frac{{2c^{2} - 20c\Delta + 23\Delta^{2} }}{18\Delta }$	$\frac{{12\left( {c - 2\Delta } \right)^{2} }}{49\Delta }$
$\pi_{L}^{BN}$	$\frac{{\left( {c + \Delta } \right)^{2} }}{9\Delta }$	$\frac{{2\left( {3c + \Delta } \right)^{2} }}{49\Delta }$
$\pi_{H}^{NB}$	$\frac{{2\left( {3c - 4\Delta } \right)^{2} }}{49\Delta }$		$\frac{{\left( {c - 2\Delta } \right)^{2} }}{9\Delta }$
$\pi_{L}^{NB}$	$\frac{{12\left( {c + \Delta } \right)^{2} }}{49\Delta }$		$\frac{{4c^{2} + 20{\text{c}}\Delta + 7\Delta^{2} }}{18\Delta }$
$\pi_{H}^{BB}$	$\frac{{11c^{2} - 26c\Delta + 17\Delta^{2} }}{18\Delta }$
$\pi_{L}^{BB}$	$\frac{{11c^{2} + 4{\text{c}}\Delta + 2\Delta^{2} }}{18\Delta }$

Now we consider the price of consumer information

For $0 \le \mu < \mu_{1}$,

$$ \begin{aligned} K_{H} & = \pi_{H}^{BN} - \pi_{H}^{NB} = \frac{{2c^{2} - 20c\Delta + 23\Delta^{2} }}{18\Delta } - \frac{{2\left( {3c - 4\Delta } \right)^{2} }}{49\Delta } = - \frac{{226c^{2} + 116c\Delta - 551\Delta^{2} }}{882\Delta }. \\ K_{L} & = \pi_{L}^{NB} - \pi_{L}^{BN} = \frac{{12\left( {c + \Delta } \right)^{2} }}{49\Delta } - \frac{{\left( {c + \Delta } \right)^{2} }}{9\Delta } = \frac{{59\left( {c + \Delta } \right)^{2} }}{441\Delta }. \\ K_{HL} & = \left( {\pi_{H}^{BB} - \pi_{H}^{NB} } \right) + \left( {\pi_{L}^{BB} - \pi_{L}^{BN} } \right) \\ & = \frac{{11c^{2} - 26c\Delta + 17\Delta^{2} }}{18\Delta } - \frac{{2\left( {3c - 4\Delta } \right)^{2} }}{49\Delta } + \frac{{11c^{2} + 4{\text{c}}\Delta + 2\Delta^{2} }}{18\Delta } - \frac{{\left( {c + \Delta } \right)^{2} }}{9\Delta } \\ & = \frac{{656c^{2} - 410c\Delta + 257\Delta^{2} }}{882\Delta }. \\ \end{aligned} $$

We can easily prove that $K_{H} > K_{HL} > K_{L}$ when $0 \le \mu < \mu_{1}$. So the information price here is ${\text{K}} = K_{H} = - \frac{{226c^{2} + 116c\Delta - 551\Delta^{2} }}{882\Delta }$.

For $\mu_{1} \le \mu < \mu_{2}$,

$$ \begin{aligned} K_{H} & = \pi_{H}^{BN} - \pi_{H}^{NB} = \frac{{12\left( {c - 2\Delta } \right)^{2} }}{49\Delta } - \frac{{2\left( {3c - 4\Delta } \right)^{2} }}{49\Delta } = \frac{{16\Delta^{2} - 6c^{2} }}{49\Delta }. \\ K_{L} & = \pi_{L}^{NB} - \pi_{L}^{BN} = \frac{{12\left( {c + \Delta } \right)^{2} }}{49\Delta } - \frac{{2\left( {3c + \Delta } \right)^{2} }}{49\Delta } = \frac{{ - 6c^{2} + 12c\Delta + 10\Delta^{2} }}{49\Delta }. \\ K_{HL} & = \left( {\pi_{H}^{BB} - \pi_{H}^{NB} } \right) + \left( {\pi_{L}^{BB} - \pi_{L}^{BN} } \right) \\ & = \frac{{11c^{2} - 26c\Delta + 17\Delta^{2} }}{18\Delta } - \frac{{2\left( {3c - 4\Delta } \right)^{2} }}{49\Delta } + \frac{{11c^{2} + 4{\text{c}}\Delta + 2\Delta^{2} }}{18\Delta } - \frac{{2\left( {3c + \Delta } \right)^{2} }}{49\Delta } \\ & = \frac{{430c^{2} - 430c\Delta + 319\Delta^{2} }}{882\Delta }. \\ \end{aligned} $$

We can easily prove that $K_{H} > K_{HL}$ and $K_{L} > K_{HL}$ when $\mu_{1} \le \mu < \mu_{2}$. Then we have

$$ K_{H} - K_{L} = \frac{{16\Delta^{2} - 6c^{2} }}{49\Delta } - \frac{{2\left( { - 3c^{2} + 6c\Delta + 5\Delta^{2} } \right)}}{49\Delta } = - \frac{6}{49}\left( {2c - \Delta } \right). $$

Thus we have $K_{H} \ge K_{L}$ when $\mu_{1} \le \mu \le \frac{{1}}{2}$ and $K_{H} < K_{L}$ when $\frac{{1}}{2} < \mu < \mu_{2}$. The respective information prices are ${\text{K}} = K_{H} = \frac{{16\Delta^{2} - 6c^{2} }}{49\Delta }$ when $\mu_{1} \le \mu \le \frac{{1}}{2}$ and ${\text{K}} = K_{L} = \frac{{ - 6c^{2} + 12c\Delta + 10\Delta^{2} }}{49\Delta }$ when $\frac{{1}}{2} < \mu < \mu_{2}$.

For $\mu_{2} \le \mu \le 1$,

$$ \begin{aligned} K_{H} & = \pi_{H}^{BN} - \pi_{H}^{NB} = \frac{{12\left( {c - 2\Delta } \right)^{2} }}{49\Delta } - \frac{{\left( {c - 2\Delta } \right)^{2} }}{9\Delta } = \frac{{59\left( {c - 2\Delta } \right)^{2} }}{441\Delta }. \\ K_{L} & = \pi_{L}^{NB} - \pi_{L}^{BN} = \frac{{4c^{2} + 20{\text{c}}\Delta + 7\Delta^{2} }}{18\Delta } - \frac{{2\left( {3c + \Delta } \right)^{2} }}{49\Delta } = \frac{{ - 128c^{2} + 764c\Delta + 307\Delta^{2} }}{882\Delta }. \\ K_{HL} & = \left( {\pi_{H}^{BB} - \pi_{H}^{NB} } \right) + \left( {\pi_{L}^{BB} - \pi_{L}^{BN} } \right) \\ & = \frac{{11c^{2} - 26c\Delta + 17\Delta^{2} }}{18\Delta } - \frac{{\left( {c - 2\Delta } \right)^{2} }}{9\Delta } + \frac{{11c^{2} + 4{\text{c}}\Delta + 2\Delta^{2} }}{18\Delta } - \frac{{2\left( {3c + \Delta } \right)^{2} }}{49\Delta }. \\ & = \frac{{656c^{2} - 902c\Delta + 503\Delta^{2} }}{882\Delta }. \\ \end{aligned} $$

We can easily prove that $K_{H} < K_{L}$ and $K_{HL} < K_{L}$ when $\mu_{2} \le \mu \le 1$. Thus the information price here is ${\text{K}} = K_{L} = \frac{{ - 128c^{2} + 764c\Delta + 307\Delta^{2} }}{882\Delta }$. Summarizing the preceding discussion, we obtain the announced result.

Proof of Lemma 1

Suppose that a strictly positive mass of old consumers pay for privacy. We assume that consumers $\theta \in \left[ {0,\theta_{L} } \right]$ pay for privacy and purchase from firm L while consumers $\theta \in \left[ {\theta_{H} ,1} \right]$ pay for privacy and purchase from firm H, $0 \le \theta_{L} \le \mu$ and $\mu \le \theta_{H} \le 1$. The functions for the maximization of the two firms’ profits are

$$ \begin{aligned} & \mathop {\text{argmax}}\limits_{{p_{H} }} \left[ {\left( {1 - \theta_{N} } \right)\left( {p_{H} - c} \right) + \mathop \int \limits_{\mu }^{{\theta_{H} }} \left( {p_{H} \left( \theta \right) - c} \right){\text{d}}\theta + \left( {1 - \theta_{H} } \right)\left( {p_{H} - c} \right)} \right]. \\ & \mathop {\text{argmax}}\limits_{{p_{L} }} [\theta_{N} p_{L} + \mathop \int \limits_{0}^{{\theta_{L} }} p_{L} {\text{d}}\theta + \mathop \int \limits_{{\theta_{L} }}^{\mu } p_{L} \left( \theta \right){\text{d}}\theta ]. \\ \end{aligned} $$

We can easily prove that both $\pi_{H}$ and $\pi_{L}$ are concave functions. By the first-order derivatives, we have $\frac{{\partial \pi_{H} }}{{p_{H} }} = \frac{{2\Delta - 2p_{H} + p_{L} + c - \Delta \theta_{H} }}{\Delta } = 0, \frac{{\partial \pi_{L} }}{{p_{L} }} = \frac{{p_{H} - 2p_{L} + \Delta \theta_{L} }}{\Delta } = 0$. Solving the two equations simultaneously, we have $p_{H} = \frac{{2c + 4\Delta - 2\theta_{H} \Delta + \theta_{L} \Delta }}{3} {\text{and }}p_{L} = \frac{{c + 2\Delta - 2\theta_{H} \Delta + 2\theta_{L} \Delta }}{3}$.

Consumers choose to pay for privacy only when they receive a higher utility, which means

$$ V + \theta q_{H} - p_{H} - c_{0} \ge V + \theta q_{H} - p_{H} \left( \theta \right) $$

and

$$ V + \theta q_{L} - p_{L} - c_{0} \ge V + \theta q_{L} - p_{L} \left( \theta \right). $$

Then we have $\frac{{2c + 4\Delta - 2\theta_{H} \Delta + \theta_{L} \Delta }}{3} + c_{0} \le \theta \Delta$ and $\frac{{c + 2\Delta - 2\theta_{H} \Delta + 2\theta_{L} \Delta }}{3} + c_{0} \le c - \theta \Delta$. The result can be expressed as $\frac{{2\mu + 4 - 2\theta_{H} + \theta_{L} }}{3} + \mu_{0} \le \theta \le \mu - \frac{{\mu + 2 - 2\theta_{H} + 2\theta_{L} }}{3} - \mu_{0}$. Comparing the lower and upper bounds of $\theta$, we find that for any $c_{0} > 0,$ there does not exist any $\theta_{H}$ and $\theta_{L} , 0 \le \theta_{L} \le \mu$ and $\mu \le \theta_{H} \le 1$, to make the inequity available. So when both firms have information and the privacy cost $c_{0} > 0$, no consumer will pay for privacy.

Proof of Proposition 5

When only firm H has consumer data, it alone has the power to price–discriminate, while consumers can pay for privacy and firm L must decide whether to set a price to retain its share of the old market.

Case 1. $\theta_{1} \left( {p_{L} } \right) \ge 0$. In this case, firm L sets a relatively low price to keep its previous market share. The function for the maximization of the two firms’ profits is

$$ \begin{aligned} & \mathop {\text{argmax}}\limits_{{p_{H} }} \left[ {\left( {1 - \theta_{N} } \right)\left( {p_{H} - c} \right) + \mathop \int \limits_{{\theta_{1} }}^{{\theta_{H}^{c} }} \left( {p_{H} \left( \theta \right) - c} \right){\text{d}}\theta + \left( {1 - \theta_{H}^{c} } \right)\left( {p_{H} - c} \right)} \right]. \\ & \mathop {\text{argmax}}\limits_{{p_{L} }} [\theta_{N} p_{L} + \mathop \int \limits_{0}^{{\theta_{1} }} p_{L} \left( \theta \right){\text{d}}\theta ]. \\ \end{aligned} $$

We can easily prove that both $\pi_{H}$ and $\pi_{L}$ are concave functions. By the first-order derivatives, we have $\frac{{\partial \pi_{H} }}{{p_{H} }} = \frac{{c + 2\Delta - 3p_{H} + 2p_{L} }}{\Delta } = 0, \frac{{\partial \pi_{L} }}{{p_{L} }} = \frac{{c + p_{H} - 4p_{L} }}{\Delta } = 0$. Solving the two equations simultaneously, we have $p_{H} = \frac{3c + 4\Delta }{5} {\text{and }}p_{L} = \frac{2c + \Delta }{5}$. To ensure the necessary condition $\theta_{1} \left( {p_{L} } \right) = \frac{{c - p_{L} }}{\Delta } \ge 0$, we have ${\text{c}} - \frac{2c + \Delta }{5} \ge 0$ and then $\mu = \frac{{c}}{\Delta } \ge \frac{{1}}{3}$. Another constraint $\theta_{H}^{c} = \frac{{p_{H} - p_{L} + c_{0} }}{\Delta } = \frac{{c + 3\Delta + 5c_{0} }}{5\Delta } \le 1$ and then we have $\mu_{0} = \frac{{c_{0} }}{\Delta } \le \frac{{2}}{5} - \frac{{1}}{5}\mu$. The firms’ profits are expressed by $\pi_{H} = \frac{{12\left( {c - 2\Delta } \right)^{2} + 25c_{0}^{2} }}{50\Delta } \,and\,\, \pi_{L} = \frac{{2\left( {2c + \Delta } \right)^{2} }}{25\Delta }$, respectively, and the total consumer surplus is given by

$$ \begin{aligned} & CS = \mathop \int \limits_{0}^{{\theta_{N} }} \left( {V + \theta q_{L} - p_{L} } \right){\text{d}}\theta + \mathop \int \limits_{{\theta_{N} }}^{1} \left( {{\text{V}} + \theta q_{H} - p_{H} } \right){\text{d}}\theta + \mathop \int \limits_{0}^{{\theta_{1} }} \left( {{\text{V}} + \theta q_{L} - p_{L} } \right){\text{d}}\theta + \mathop \int \limits_{{\theta_{1} }}^{{\theta_{H}^{c} }} \left( {{\text{V}} + \theta q_{H} - p_{H} \left( \theta \right)} \right){\text{d}}\theta + \mathop \int \limits_{{\theta_{H}^{c} }}^{1} \left( {{\text{V}} + \theta q_{H} - p_{H} } \right){\text{d}}\theta \\ & \quad = \frac{{2c^{2} + 25c_{0}^{2} + 10c_{0} \left( {c - 2q_{H} + 2q_{L} } \right)}}{{50\left( {q_{H} - q_{L} } \right)}} - \frac{{1}}{25}\left( {24c + 6q_{H} - 31q_{L} } \right) + 2V. \\ \end{aligned} $$

Case 2. $\theta_{1} \left( {p_{L} } \right) < 0$. In this case, firm L chooses to relinquish its share of the old market. The profit maximization functions are

$$ \begin{aligned} & \mathop {\text{argmax}}\limits_{{p_{H} }} \left[ {\left( {1 - \theta_{N} } \right)\left( {p_{H} - c} \right) + \mathop \int \limits_{0}^{{\theta_{H}^{c} }} \left( {p_{H} \left( \theta \right) - c} \right){\text{d}}\theta + \left( {1 - \theta_{H}^{c} } \right)\left( {p_{H} - c} \right)} \right]. \\ & \mathop {\text{argmax}}\limits_{{p_{L} }} \theta_{N} p_{L} . \\ \end{aligned} $$

It is obvious that both $\pi_{H}$ and $\pi_{L}$ are concave functions. By the first-order derivatives, we have $\frac{{\partial \pi_{H} }}{{p_{H} }} = \frac{{c + 2\Delta - 3p_{H} + 2p_{L} }}{\Delta } = 0, \frac{{\partial \pi_{L} }}{{p_{L} }} = \frac{{p_{H} - 2p_{L} }}{\Delta } = 0$. Solving the two equations simultaneously, we have $p_{H} = \frac{c + 2\Delta }{2} {\text{and }}p_{L} = \frac{c + 2\Delta }{4}$. To ensure the necessary condition $\theta_{1} \left( {p_{L} } \right) = \frac{{c - p_{L} }}{\Delta } < 0$, we have ${\text{c}} - \frac{c + 2\Delta }{4} < 0$ and then $\mu = \frac{{c}}{\Delta } < \frac{{2}}{3}$. Another constraint $\theta_{H}^{c} = \frac{{p_{H} - p_{L} + c_{0} }}{\Delta } = \frac{{c + 2\Delta + 4c_{0} }}{4\Delta } \le 1$ and then we have $\mu_{0} = \frac{{c_{0} }}{\Delta } \le \frac{{1}}{2} - \frac{{1}}{4}\mu$. The firms’ profits are expressed respectively by $\pi_{H} = \frac{{3c^{2} - 36c\Delta + 44\Delta^{2} + 16c_{0}^{2} }}{32\Delta } \,and\,\, \pi_{L} = \frac{{\left( {c + 2\Delta } \right)^{2} }}{16\Delta }$, and the total consumer surplus is given by

$$ \begin{aligned} & CS = \mathop \int \limits_{0}^{{\theta_{N} }} \left( {V + \theta q_{L} - p_{L} } \right){\text{d}}\theta + \mathop \int \limits_{{\theta_{N} }}^{1} \left( {{\text{V}} + \theta q_{H} - p_{H} } \right){\text{d}}\theta + \mathop \int \limits_{0}^{{\theta_{H}^{c} }} \left( {{\text{V}} + \theta q_{H} - p_{H} \left( \theta \right)} \right){\text{d}}\theta + \mathop \int \limits_{{\theta_{H}^{c} }}^{1} \left( {{\text{V}} + \theta q_{H} - p_{H} } \right){\text{d}}\theta \\ & \quad = \frac{{c^{2} + 8c_{0}^{2} + 4c_{0} \left( {c - 2q_{H} + 2q_{L} } \right)}}{{16\left( {q_{H} - q_{L} } \right)}} + \frac{{1}}{4}\left( { - 3c - 3q_{H} + 7q_{L} } \right) + 2V. \\ \end{aligned} $$

From the preceding discussion, we know that when $\mu \ge \frac{{2}}{3}$, only case 1 is available and when $\mu < \frac{{1}}{3}$, only case 2 is available. When $\frac{{1}}{3} \le \mu < \frac{{2}}{3}$, firm L can choose between the two cases. Comparing them, we calculate

$$ \begin{aligned} & \pi_{L}^{case1} - \pi_{L}^{case2} = \frac{{2\left( {2c + \Delta } \right)^{2} }}{25\Delta } - \frac{{\left( {c + 2\Delta } \right)^{2} }}{16\Delta } = \frac{{103c^{2} + 28c\Delta - 68\Delta^{2} }}{400\Delta } \\ & \quad = \frac{\Delta }{400}\left( {103\mu^{2} + 28\mu - 68} \right). \\ \end{aligned} $$

We solve the inequality $103\mu^{2} + 28\mu - 68 \ge 0$ which is due to $\mu \in \left[ {\frac{{1}}{3},\frac{{2}}{3}} \right)$. We find that $\pi_{L}^{case1} < \pi_{L}^{case2}$ is always true when $\frac{{1}}{3} \le \mu < \frac{{2}}{3}$. At the same time, the upper bound of $\mu_{0}$ also satisfies $\frac{{1}}{2} - \frac{{1}}{4}\mu > \frac{{2}}{5} - \frac{{1}}{5}\mu$. Ultimately, the preceding conditions can be expressed as follows: when $0 \le \mu < \frac{{2}}{3}$, the equilibrium operates as case 2; when $\frac{{2}}{3} \le \mu \le 1$, the equilibrium operates as case 1. Thus we obtain the announced result.

Proof of Proposition 6

When only firm L has information, this firm have the exclusive power to price- discriminate, while consumers can pay for privacy and firm H must decide whether to set a price to retain a share of the old market.

Case 1. $\theta_{2} \left( {p_{H} } \right) \le 1.$ In this case, firm H sets a relatively low price to hold on to its market share. The profit maximization functions are

$$ \begin{aligned} & \mathop {\text{argmax}}\limits_{{p_{H} }} \left[ {\left( {1 - \theta_{N} } \right)\left( {p_{H} - c} \right) + \left( {1 - \theta_{2} } \right)\left( {p_{H} - c} \right)} \right]. \\ & \mathop {\text{argmax}}\limits_{{p_{L} }} [\theta_{N} p_{L} + \mathop \int \limits_{0}^{{\theta_{L}^{c} }} p_{L} {\text{d}}\theta + \mathop \int \limits_{{\theta_{L}^{c} }}^{{\theta_{2} }} p_{L} \left( \theta \right){\text{d}}\theta ]. \\ \end{aligned} $$

It can readily be proved that both $\pi_{H}$ and $\pi_{L}$ are concave functions. By the first-order derivatives, we have $\frac{{\partial \pi_{H} }}{{p_{H} }} = \frac{{2\left( {c + \Delta } \right) - 4p_{H} + p_{L} }}{\Delta } = 0, \frac{{\partial \pi_{L} }}{{p_{L} }} = \frac{{2p_{H} - 3p_{L} }}{\Delta } = 0$. Solving the two equations simultaneously, we have $p_{H} = \frac{{3\left( {c + \Delta } \right)}}{5} {\text{and }}p_{L} = \frac{{3\left( {c + \Delta } \right)}}{5}$. To ensure the necessary condition $\theta_{2} \left( {p_{H} } \right) = \frac{{p_{H} }}{\Delta } \le 1$, we have $\frac{{3\left( {c + \Delta } \right)}}{5} - \Delta \le 0$ and then $\mu = \frac{{c}}{\Delta } \le \frac{{2}}{3}$. Another constraint $\theta_{L}^{c} = \frac{{p_{H} - p_{L} - c_{0} }}{\Delta } = \frac{{c + \Delta - 5c_{0} }}{5\Delta } \ge 0$ and then we have $\mu_{0} = \frac{{c_{0} }}{\Delta } \le \frac{{1}}{5} + \frac{{1}}{5}\mu$. The firms profits are expressed by $\pi_{H} = \frac{{2\left( {2c - 3\Delta } \right)^{2} }}{25\Delta } and \pi_{L} = \frac{{12\left( {c + \Delta } \right)^{2} + 25c_{0}^{2} }}{50\Delta }$, respectively, and the total consumer surplus is given by

$$ \begin{aligned} & CS = \mathop \int \limits_{0}^{{\theta_{N} }} \left( {V + \theta q_{L} - p_{L} } \right){\text{d}}\theta + \mathop \int \limits_{{\theta_{N} }}^{1} \left( {{\text{V}} + \theta q_{H} - p_{H} } \right){\text{d}}\theta + \mathop \int \limits_{0}^{{\theta_{L}^{c} }} \left( {{\text{V}} + \theta q_{L} - p_{L} } \right){\text{d}}\theta + \mathop \int \limits_{{\theta_{L}^{c} }}^{{\theta_{2} }} \left( {V + \theta q_{L} - p_{L} \left( \theta \right)} \right){\text{d}}\theta + \mathop \int \limits_{{\theta_{2} }}^{1} \left( {{\text{V}} + \theta q_{H} - p_{H} } \right){\text{d}}\theta \\ & \quad = \frac{{2c^{2} + 25c_{0}^{2} - 10c_{0} \left( {c + q_{H} - q_{L} } \right)}}{{50\left( {q_{H} - q_{L} } \right)}} - \frac{{1}}{25}\left( {28c + 4q_{H} - 29q_{L} } \right) + 2V. \\ \end{aligned} $$

Case 2. $\theta_{2} \left( {p_{H} } \right) > 1.$ In this case, firm H chooses to relinquish its previous share of the old market. The profit maximization functions are

$$ \begin{aligned} & \mathop {\text{argmax}}\limits_{{p_{H} }} \left( {1 - \theta_{N} } \right)\left( {p_{H} - c} \right). \\ & \mathop {\text{argmax}}\limits_{{p_{L} }} [\theta_{N} p_{L} + \mathop \int \limits_{0}^{{\theta_{L}^{c} }} p_{L} {\text{d}}\theta + \mathop \int \limits_{{\theta_{L}^{c} }}^{1} p_{L} \left( \theta \right){\text{d}}\theta ]. \\ \end{aligned} $$

We can easily prove that both $\pi_{H}$ and $\pi_{L}$ are concave functions. By the first-order derivatives, we have $\frac{{\partial \pi_{H} }}{{p_{H} }} = \frac{{c + \Delta - 2p_{H} + p_{L} }}{\Delta } = , \frac{{\partial \pi_{L} }}{{p_{L} }} = \frac{{2p_{H} - 3p_{L} }}{\Delta } = 0$. Solving the two equations simultaneously, we have $p_{H} = \frac{{3\left( {c + \Delta } \right)}}{4} {\text{and }}p_{L} = \frac{c + \Delta }{2}$. To ensure the necessary condition $\theta_{2} \left( {p_{H} } \right) = \frac{{p_{H} }}{\Delta } > 1$, we have $\frac{{3\left( {c + \Delta } \right)}}{4\Delta } - 1 > 0$ and then $\mu = \frac{{c}}{\Delta } > \frac{{1}}{3}$. Another constraint $\theta_{L}^{c} = \frac{{p_{H} - p_{L} - c_{0} }}{\Delta } = \frac{{c + \Delta - 4c_{0} }}{4\Delta } \ge 0$ and then we have $\mu_{0} = \frac{{c_{0} }}{\Delta } \le \frac{{1}}{4} + \frac{{1}}{4}\mu$. The firms’ profits are expressed by $\pi_{H} = \frac{{\left( {c - 3\Delta } \right)^{2} }}{16\Delta } and \pi_{L} = \frac{{3c^{2} + 30c\Delta + 11\Delta^{2} + 16c_{0}^{2} }}{32\Delta }$, and the total consumer surplus is given by

$$ \begin{aligned} & CS = \mathop \int \limits_{0}^{{\theta_{N} }} \left( {V + \theta q_{L} - p_{L} } \right){\text{d}}\theta + \mathop \int \limits_{{\theta_{N} }}^{1} \left( {{\text{V}} + \theta q_{H} - p_{H} } \right){\text{d}}\theta + \mathop \int \limits_{0}^{{\theta_{L}^{c} }} \left( {{\text{V}} + \theta q_{L} - p_{L} } \right){\text{d}}\theta + \mathop \int \limits_{{\theta_{L}^{c} }}^{1} \left( {{\text{V}} + \theta q_{L} - p_{L} \left( \theta \right)} \right){\text{d}}\theta \\ & \quad = \frac{{c^{2} + 8c_{0}^{2} - 4c_{0} \left( {c + q_{H} - q_{L} } \right)}}{{16\left( {q_{H} - q_{L} } \right)}} - \frac{{1}}{16}\left( {22c + 7q_{H} - 23q_{L} } \right) + 2V. \\ \end{aligned} $$

From the preceding discussion, we know that when $\mu \le \frac{{1}}{3}$, only case 1 is available; and when $\mu > \frac{{2}}{3}$, only case 2 is available. When $\frac{{1}}{3} < \mu \le \frac{{2}}{3}$, firm H can choose between the two cases. Comparing them, we calculate

$$ \pi_{H}^{case1} - \pi_{H}^{case2} = \frac{{2\left( {2c - 3\Delta } \right)^{2} }}{25\Delta } - \frac{{\left( {c - 3\Delta } \right)^{2} }}{16\Delta } = \frac{{103c^{2} - 234c\Delta + 63\Delta^{2} }}{400\Delta }. $$

We solve the inequality $113\mu^{2} + 234\mu + 63 \ge 0$ which is due to $\mu \in \left[ {\frac{{1}}{3},\frac{{2}}{3}} \right]$. The result shows that $\pi_{H}^{case1} - \pi_{H}^{case2} < 0$ is always true for $\frac{{1}}{3} < \mu \le \frac{{2}}{3}$. At the same time, the upper bound of $\mu_{0}$ also satisfies $\frac{{1}}{4} + \frac{{1}}{4}\mu > \frac{{1}}{5} + \frac{{1}}{5}\mu$. Ultimately, the preceding conditions can be expressed as follows: when $0 \le \mu \le \frac{{1}}{3}$, the equilibrium operates as case 1; when $\frac{{1}}{3} < \mu \le 1$, the equilibrium operates as case 2. Thus we obtain the announced result.

Proof of Proposition 7

Through the preceding analysis, we have determined the profits of firm H and firm L profits for the different cases in which consumers can pay for privacy. These are shown in the following charts.

• (1) In region 1, the customers of both firms can pay for privacy, and we have

$\mu$	$0 \le \mu < \frac{{1}}{3}$	$\frac{{1}}{3} \le \mu < \frac{{2}}{3}$	$\frac{{2}}{3} \le \mu \le 1$
$\pi_{H}^{BN}$	$\frac{{3c^{2} - 36c\Delta + 44\Delta^{2} + 16c_{0}^{2} }}{32\Delta }$		$\frac{{12\left( {c - 2\Delta } \right)^{2} + 25c_{0}^{2} }}{50\Delta }$
$\pi_{L}^{BN}$	$\frac{{\left( {c + 2\Delta } \right)^{2} }}{16\Delta }$		$\frac{{2\left( {2c + \Delta } \right)^{2} }}{25\Delta }$
$\pi_{H}^{NB}$	$\frac{{2\left( {2c - 3\Delta } \right)^{2} }}{25\Delta }$	$\frac{{\left( {c - 3\Delta } \right)^{2} }}{16\Delta }$
$\pi_{L}^{NB}$	$\frac{{12\left( {c + \Delta } \right)^{2} + 25c_{0}^{2} }}{50\Delta }$	$\frac{{3c^{2} + 30c\Delta + 11\Delta^{2} + 16c_{0}^{2} }}{32\Delta }$
$\pi_{H}^{BB}$	$\frac{{11c^{2} - 26c\Delta + 17\Delta^{2} }}{18\Delta }$
$\pi_{L}^{BB}$	$\frac{{11c^{2} + 4{\text{c}}\Delta + 2\Delta^{2} }}{18\Delta }$

Now we consider the price of consumer information.

For $0 \le \mu < \frac{{1}}{3}$

$$ \begin{aligned} K_{H} & = \pi_{H}^{BN} - \pi_{H}^{NB} = \frac{{3c^{2} - 36c\Delta + 44\Delta^{2} + 16c_{0}^{2} }}{32\Delta } - \frac{{2\left( {2c - 3\Delta } \right)^{2} }}{25\Delta } \\ & = \frac{{ - 181c^{2} - 132c\Delta + 524\Delta^{2} + 400c_{0}^{2} }}{800\Delta }. \\ K_{L} & = \pi_{L}^{NB} - \pi_{L}^{BN} = \frac{{12\left( {c + \Delta } \right)^{2} + 25c_{0}^{2} }}{50\Delta } - \frac{{\left( {c + 2\Delta } \right)^{2} }}{16\Delta } = \frac{{c^{2} + 22c\Delta + 3\Delta^{2} + 16c_{0}^{2} }}{32\Delta }. \\ K_{HL} & = \left( {\pi_{H}^{BB} - \pi_{H}^{NB} } \right) + \left( {\pi_{L}^{BB} - \pi_{L}^{BN} } \right) \\ & = \frac{{11c^{2} - 26c\Delta + 17\Delta^{2} }}{18\Delta } - \frac{{2\left( {2c - 3\Delta } \right)^{2} }}{25\Delta } + \frac{{11c^{2} + 4{\text{c}}\Delta + 2\Delta^{2} }}{18\Delta } - \frac{{\left( {c + 2\Delta } \right)^{2} }}{16\Delta } \\ & = \frac{{3023c^{2} - 1844c\Delta + 308\Delta^{2} }}{3600\Delta }. \\ \end{aligned} $$

We can easily prove that $K_{H} > K_{HL}$ and $K_{H} > K_{L}$ for any $c_{0} > 0$ when $0 \le \mu < \frac{{1}}{3}$. So the information price here is ${\text{K}} = K_{H} = \frac{{ - 181c^{2} - 132c\Delta + 524\Delta^{2} + 400c_{0}^{2} }}{800\Delta }$.

For $\frac{{1}}{3} \le \mu < \frac{{2}}{3}$

$$ \begin{aligned} K_{H} & = \pi_{H}^{BN} - \pi_{H}^{NB} = \frac{{3c^{2} - 36c\Delta + 44\Delta^{2} + 16c_{0}^{2} }}{32\Delta } - \frac{{\left( {c - 3\Delta } \right)^{2} }}{16\Delta } \\ & = \frac{{c^{2} - 24c\Delta + 26\Delta^{2} + 16c_{0}^{2} }}{32\Delta }. \\ K_{L} & = \pi_{L}^{NB} - \pi_{L}^{BN} = \frac{{3c^{2} + 30c\Delta + 11\Delta^{2} + 16c_{0}^{2} }}{32\Delta } - \frac{{\left( {c + 2\Delta } \right)^{2} }}{16\Delta } \\ & = \frac{{c^{2} + 22c\Delta + 3\Delta^{2} + 16c_{0}^{2} }}{32\Delta }. \\ K_{HL} & = \left( {\pi_{H}^{BB} - \pi_{H}^{NB} } \right) + \left( {\pi_{L}^{BB} - \pi_{L}^{BN} } \right) \\ & = \frac{{11c^{2} - 26c\Delta + 17\Delta^{2} }}{18\Delta } - \frac{{\left( {c - 3\Delta } \right)^{2} }}{16\Delta } + \frac{{11c^{2} + 4{\text{c}}\Delta + 2\Delta^{2} }}{18\Delta } - \frac{{\left( {c + 2\Delta } \right)^{2} }}{16\Delta } \\ & = \frac{{1}}{144}\left( { - 158c + \frac{{158c^{2} }}{\Delta } + 35\Delta } \right). \\ \end{aligned} $$

We can easily prove that $K_{H} > K_{HL}$ and $K_{L} > K_{HL}$ for any $c_{0} > 0$ when $\frac{{1}}{3} \le \mu < \frac{{2}}{3}$. And we have

$$ K_{H} - K_{L} = \frac{23}{32}\left( { - 2c + \Delta } \right) = \frac{23\Delta }{32}\left( { - 2\mu + 1} \right). $$

Thus we have $K_{H} \ge K_{L}$ when $\frac{{1}}{3} \le \mu \le \frac{{1}}{2}$ and $K_{H} < K_{L}$ when $\frac{{1}}{2} < \mu < \frac{{2}}{3}$. The information prices here are ${\text{K}} = K_{H} = \frac{{c^{2} - 24c\Delta + 26\Delta^{2} + 16c_{0}^{2} }}{32\Delta }$ when $\frac{{1}}{3} \le \mu \le \frac{{1}}{2}$ and ${\text{K}} = K_{L} = \frac{{71c^{2} + 92c\Delta - 4\Delta^{2} + 200c_{0}^{2} }}{400\Delta }$ when $\frac{{1}}{2} < \mu \le 1$.

For $\frac{{2}}{3} \le \mu \le 1$

$$ \begin{aligned} K_{H} & = \pi_{H}^{BN} - \pi_{H}^{NB} = \frac{{12\left( {c - 2\Delta } \right)^{2} + 25c_{0}^{2} }}{50\Delta } - \frac{{\left( {c - 3\Delta } \right)^{2} }}{16\Delta } \\ & = \frac{{71c^{2} - 234c\Delta + 159\Delta^{2} + 200c_{0}^{2} }}{400\Delta }. \\ K_{L} & = \pi_{L}^{NB} - \pi_{L}^{BN} = \frac{{3c^{2} + 30c\Delta + 11\Delta^{2} + 16c_{0}^{2} }}{32\Delta } - \frac{{2\left( {2c + \Delta } \right)^{2} }}{25\Delta } \\ & = \frac{{ - 181c^{2} + 494c\Delta + 211\Delta^{2} + 400c_{0}^{2} }}{800\Delta }. \\ K_{HL} & = \left( {\pi_{H}^{BB} - \pi_{H}^{NB} } \right) + \left( {\pi_{L}^{BB} - \pi_{L}^{BN} } \right) \\ & = \frac{{11c^{2} - 26c\Delta + 17\Delta^{2} }}{18\Delta } - \frac{{2\left( {2c + \Delta } \right)^{2} }}{25\Delta } + \frac{{11c^{2} + 4{\text{c}}\Delta + 2\Delta^{2} }}{18\Delta } - \frac{{\left( {c - 3\Delta } \right)^{2} }}{16\Delta } \\ & = \frac{{3023c^{2} - 4202c\Delta + 1487\Delta^{2} }}{3600\Delta }. \\ \end{aligned} $$

We can easily prove that $K_{H} < K_{L}$ and $K_{HL} < K_{L}$ for any $c_{0} > 0$ when $\frac{{2}}{3} \le \mu \le 1$. Thus the information price here is ${\text{K}} = K_{L} = \frac{{ - 181c^{2} + 494c\Delta + 211\Delta^{2} + 400c_{0}^{2} }}{800\Delta }$.

• (2) In regions 2 and 3, only exclusive firm’s consumers may purchase product and pay for privacy, then we have

$\mu$	$0 \le \mu < \frac{{1}}{2}$	$\frac{{1}}{2} < \mu \le 1$
$\pi_{H}^{BN}$	$\frac{{3c^{2} - 36c\Delta + 44\Delta^{2} + 16c_{0}^{2} }}{32\Delta }$	$\frac{{12\left( {c - 2\Delta } \right)^{2} }}{49\Delta }$
$\pi_{L}^{BN}$	$\frac{{\left( {c + 2\Delta } \right)^{2} }}{16\Delta }$	$\frac{{2\left( {3c + \Delta } \right)^{2} }}{49\Delta }$
$\pi_{H}^{NB}$	$\frac{{2\left( {3c - 4\Delta } \right)^{2} }}{49\Delta }$	$\frac{{\left( {c - 3\Delta } \right)^{2} }}{16\Delta }$
$\pi_{L}^{NB}$	$\frac{{12\left( {c + \Delta } \right)^{2} }}{49\Delta }$	$\frac{{3c^{2} + 30c\Delta + 11\Delta^{2} + 16c_{0}^{2} }}{32\Delta }$
$\pi_{H}^{BB}$	$\frac{{11c^{2} - 26c\Delta + 17\Delta^{2} }}{18\Delta }$
$\pi_{L}^{BB}$	$\frac{{11c^{2} + 4{\text{c}}\Delta + 2\Delta^{2} }}{18\Delta }$

Now we consider the price of consumer information.

For $0 \le \mu < \frac{{1}}{2}$,

$$ \begin{aligned} K_{H} & = \pi_{H}^{BN} - \pi_{H}^{NB} = \frac{{3c^{2} - 36c\Delta + 44\Delta^{2} + 16c_{0}^{2} }}{32\Delta } - \frac{{2\left( {3c - 4\Delta } \right)^{2} }}{49\Delta } \\ & = \frac{{ - 429c^{2} - 228c\Delta + 1132\Delta^{2} + 784c_{0}^{2} }}{1568\Delta }. \\ K_{L} & = \pi_{L}^{NB} - \pi_{L}^{BN} = \frac{{12\left( {c + \Delta } \right)^{2} }}{49\Delta } - \frac{{\left( {c + 2\Delta } \right)^{2} }}{16\Delta } = \frac{{143c^{2} + 188c\Delta - 4\Delta^{2} }}{784\Delta }. \\ K_{HL} & = \left( {\pi_{H}^{BB} - \pi_{H}^{NB} } \right) + \left( {\pi_{L}^{BB} - \pi_{L}^{BN} } \right) \\ & = \frac{{11c^{2} - 26c\Delta + 17\Delta^{2} }}{18\Delta } - \frac{{2\left( {3c - 4\Delta } \right)^{2} }}{49\Delta } + \frac{{11c^{2} + 4{\text{c}}\Delta + 2\Delta^{2} }}{18\Delta } - \frac{{\left( {c + 2\Delta } \right)^{2} }}{16\Delta } \\ & = \frac{{5591c^{2} - 3476c\Delta + 1076\Delta^{2} }}{7056\Delta }. \\ \end{aligned} $$

We can easily prove that $K_{H} > K_{L}$ and $K_{H} > K_{HL}$ for any $c_{0} > 0$ when $0 \le \mu \le \frac{{1}}{2}$. Thus the information price here is ${\text{K}} = K_{H} = \frac{{ - 429c^{2} - 228c\Delta + 1132\Delta^{2} + 784c_{0}^{2} }}{1568\Delta }$.

For $\frac{{1}}{2} < \mu \le 1$,

$$ \begin{aligned} K_{H} & = \pi_{H}^{BN} - \pi_{H}^{NB} = \frac{{12\left( {c - 2\Delta } \right)^{2} }}{49\Delta } - \frac{{\left( {c - 3\Delta } \right)^{2} }}{16\Delta } = \frac{{143c^{2} - 474c\Delta + 327\Delta^{2} }}{784\Delta }. \\ K_{L} & = \pi_{L}^{NB} - \pi_{L}^{BN} = \frac{{3c^{2} + 30c\Delta + 11\Delta^{2} + 16c_{0}^{2} }}{32\Delta } - \frac{{2\left( {3c + \Delta } \right)^{2} }}{49\Delta } \\ & = \frac{{ - 429c^{2} + 1086c\Delta + 475\Delta^{2} + 784c_{0}^{2} }}{1568\Delta }. \\ K_{HL} & = \left( {\pi_{H}^{BB} - \pi_{H}^{NB} } \right) + \left( {\pi_{L}^{BB} - \pi_{L}^{BN} } \right) \\ & = \frac{{11c^{2} - 26c\Delta + 17\Delta^{2} }}{18\Delta } - \frac{{\left( {c - 3\Delta } \right)^{2} }}{16\Delta } + \frac{{11c^{2} + 4{\text{c}}\Delta + 2\Delta^{2} }}{18\Delta } - \frac{{2\left( {3c + \Delta } \right)^{2} }}{49\Delta } \\ & = \frac{{5591c^{2} - 7706c\Delta + 3191\Delta^{2} }}{7056\Delta }. \\ \end{aligned} $$

We can easily prove that $K_{H} < K_{L}$ and $K_{L} > K_{HL}$ for any $c_{0} > 0$ when $\frac{{1}}{2} < \mu \le 1$. Thus the information price here is ${\text{K}} = K_{L} = \frac{{ - 429c^{2} + 1086c\Delta + 475\Delta^{2} + 784c_{0}^{2} }}{1568\Delta }$. Summarizing the discussion above, we obtain the announced Proposition 7.

Proof of Corollary 2

In the BN case, only firm H has the information and consumers pay for privacy. We know firm H’s profits

$$ \pi_{H} = \left\{ {\begin{array}{*{20}l} {\frac{{3c^{2} - 36c\Delta + 44\Delta^{2} + 16c_{0}^{2} }}{32\Delta },} \hfill & {0 \le \mu < \frac{{2}}{3}\, and \,0 < \mu_{0} \le \frac{{1}}{2} - \frac{{1}}{4}\mu } \hfill \\ {\frac{{12\left( {c - 2\Delta } \right)^{2} + 25c_{0}^{2} }}{50\Delta },} \hfill & {\frac{{2}}{3} \le \mu \le 1 \,and\, 0 < \mu_{0} \le \frac{{2}}{5} - \frac{{1}}{5}\mu } \hfill \\ \end{array} } \right.. $$

While firm L’s profits are not affected by the value of $c_{0}$. For any $c_{0} > 0$, we can easily determine that $\pi_{H}^{'} \left( {c_{0} } \right) > 0$ which is due to $0 \le \mu \le 1$. That means $\pi_{H}$ always increases with $c_{0}$. When $c_{0}$ exceeds the upper bound, the profits are the same as in the no-privacy case (i.e., they are linear).

In the NB case, only firm L has the information and consumers can pay for privacy. We know firm L’s profits

$$ \pi_{H} = \left\{ {\begin{array}{*{20}l} {\frac{{12\left( {c + \Delta } \right)^{2} + 25c_{0}^{2} }}{50\Delta },} \hfill & {0 \le \mu \le \frac{{1}}{3}\, and\, 0 < \mu_{0} \le \frac{{1}}{5} + \frac{{1}}{5}\mu } \hfill \\ {\frac{{3c^{2} + 30c\Delta + 11\Delta^{2} + 16c_{0}^{2} }}{32\Delta },} \hfill & {\frac{{1}}{3} < \mu \le 1 \,and \,0 < \mu_{0} \le \frac{{1}}{4} + \frac{{1}}{4}\mu } \hfill \\ \end{array} } \right., $$

while firm H’s profits are not related to $c_{0}$. For any $c_{0} > 0$, we can easily determine that $\pi_{L}^{'} \left( {c_{0} } \right) > 0$ which is due to $0 \le \mu \le 1$. That means $\pi_{L}$ also increases with $c_{0}$. As $\mu_{0}$ exceeds the upper bound, the profits are the same as in the no-privacy case. Thus we obtain the announced Corollary 2.

Proof of Corollary 3

In the BN case, when $0 \le \mu < \frac{{2}}{3}$, we have ${\text{CS}}^{'} \left( {c_{0} } \right) = \frac{{16c_{0} + 4\left( {c - 2q_{H} + 2q_{L} } \right)}}{{16\left( {q_{H} - q_{L} } \right)}} = \mu_{0} + \frac{{1}}{4}\mu - \frac{{1}}{2} < 0$ for any $\mu_{0} \in \left( {0,\frac{{1}}{2} - \frac{{1}}{4}\mu } \right)$, which means that ${\text{CS}}$ decreases as $c_{0}$ rises. When $\frac{{2}}{3} \le \mu \le 1$, ${\text{CS}}^{'} \left( {c_{0} } \right) = \frac{{50c_{0} + 10\left( {c - 2q_{H} + 2q_{L} } \right)}}{{50\left( {q_{H} - q_{L} } \right)}} = \mu_{0} + \frac{{1}}{5}\mu - \frac{{2}}{5} < 0$ for any $\mu_{0} \in \left( {0,\frac{{2}}{5} - \frac{{1}}{5}\mu } \right)$, which means that here too ${\text{CS}}$ decreases as $c_{0}$ increases.

In the NB case, when $0 \le \mu \le \frac{{1}}{3}$, we have ${\text{CS}}^{'} \left( {c_{0} } \right) = \frac{{50c_{0} - 10\left( {c + q_{H} - q_{L} } \right)}}{{50\left( {q_{H} - q_{L} } \right)}} = \mu_{0} - \frac{{1}}{5}\mu - \frac{{1}}{5} < 0$ for any $\mu_{0} \in \left( {0,\frac{{1}}{5} + \frac{{1}}{5}\mu } \right)$, which means that ${\text{CS}}$ decreases with rising $c_{0}$. Likewise, when $\frac{{1}}{3} < \mu \le 1$, ${\text{CS}}^{'} \left( {c_{0} } \right) = \frac{{16c_{0} - 4\left( {c + q_{H} - q_{L} } \right)}}{{16\left( {q_{H} - q_{L} } \right)}} = \mu_{0} - \frac{{1}}{4}\mu - \frac{{1}}{4} < 0$ for any $\mu_{0} \in \left( {0,\frac{{1}}{4} + \frac{{1}}{4}\mu } \right)$, which means that once again ${\text{CS}}$ falls as $c_{0}$ rises.

Proof of Corollary 4

From the discussion in Proposition 7,we have the information prices in different regions, as summarized in Tables 1 and 2.

Table 1 The information price in different regions

Table 2 The information price in different regions

${\text{K}}^{\prime } \left( {c_{0} } \right) > 0$ is always available for any $c_{0} > 0$ in whichever region. At the same time, the information prices in region II and III are always larger than the corresponding prices in region I. So we can say that the information prices offered by the DS always increase with $c_{0} ,$ and we arrive at the announced results.

Proof of Proposition 8

(1) when $0 < \mu \le \mu_{1} , 0 < \mu_{0} < \frac{{1}}{2} - \frac{{1}}{4}\mu$, the difference in industry profits with and without transparency is given by

$$ \begin{aligned} \pi^{D} &= \pi^{with} - \pi^{without} \\ & = \frac{{3c^{2} - 36c\Delta + 44\Delta^{2} + 16c_{0}^{2} }}{32\Delta } + \frac{{\left( {c + 2\Delta } \right)^{2} }}{16\Delta } - \frac{{1}}{18}\left( { - 20c + \frac{{2c^{2} }}{\Delta } + 23\Delta } \right) - \frac{{\left( {c + \Delta } \right)^{2} }}{9\Delta } \\ & = \frac{{ - 19c^{2} + 4c\Delta + 68\Delta^{2} + 144c_{0}^{2} }}{288\Delta } > 0. \\ \end{aligned} $$

The consumer surplus difference is

$$ \begin{aligned} CS^{D} &= CS^{with} - CS^{without} \\ &= \frac{{72c_{0}^{2} + 36c_{0} \left( {c - 2\Delta } \right) + \left( {c + 22\Delta } \right)\left( {c - 2\Delta } \right)}}{144\Delta } \\ &= \frac{\Delta }{144}\left[ {72\mu_{0}^{2} + \left( {36\mu_{0} + \mu + 22} \right)\left( {\mu - 2} \right)} \right] < 0. \\ \end{aligned} $$

The social welfare difference is

$$ \begin{aligned} SW^{D} &= \pi^{D} + CS^{D} \\ & = \frac{{ - 17c^{2} + 44c\Delta - 20\Delta^{2} + 72cc_{0} - 144\Delta c_{0} + 288c_{0}^{2} }}{288\Delta } \\ & = \frac{\Delta }{288}\left( { - 17\mu^{2} + 44\mu - 20} \right) + \Delta \left[ {\mu_{0}^{2} - \left( {\frac{{1}}{2} - \frac{{1}}{4}\mu } \right)\mu_{0} } \right]. \\ \end{aligned} $$

We can easily find that $- 17\mu^{2} + 44\mu - 20 < 0$ and $\mu_{0}^{2} - \left( {\frac{{1}}{2} - \frac{{1}}{4}\mu } \right)\mu_{0} \le 0$ for any $0 < \mu \le \mu_{1} , 0 < \mu_{0} < \frac{{1}}{2} - \frac{{1}}{4}\mu$, so we have $SW^{D} < 0$.

• (2) When $\mu_{1} < \mu \le \frac{{2}}{3}, 0 < \mu_{0} < \frac{{1}}{2} - \frac{{1}}{4}\mu$, the difference in industry profits with and without transparency is given by

$$ \begin{aligned} \pi^{D} &= \pi^{with} - \pi^{without} \\ & = \frac{{3c^{2} - 36c\Delta + 44\Delta^{2} + 16c_{0}^{2} }}{32\Delta } + \frac{{\left( {c + 2\Delta } \right)^{2} }}{16\Delta } - \frac{{12\left( {c - 2\Delta } \right)^{2} }}{49\Delta } - \frac{{2\left( {3c + \Delta } \right)^{2} }}{49\Delta } \\ & = \frac{{ - 715c^{2} - 220c\Delta + 948\Delta^{2} + 784c_{0}^{2} }}{1568\Delta } > 0. \\ \end{aligned} $$

The consumer surplus difference is

$$ \begin{aligned} CS^{D} &= CS^{with} - CS^{without} \\ & = \frac{{392c_{0}^{2} + 196c_{0} \left( {c - 2\Delta } \right) + \left( {17c + 246\Delta } \right)\left( {c - 2\Delta } \right)}}{784\Delta } \\ & = \frac{\Delta }{784}\left[ {392\mu_{0}^{2} + \left( {196\mu_{0} + 17\mu + 246} \right)\left( {\mu - 2} \right)} \right] < 0. \\ \end{aligned} $$

The social welfare difference is

$$ \begin{aligned} SW^{D} &= \pi^{D} + S^{D} \\ & = \frac{{ - 681c^{2} + 204c\Delta - 36\Delta^{2} + 392cc_{0} - 784\Delta c_{0} + 1568c_{0}^{2} }}{1568\Delta } \\ & = \frac{\Delta }{1568}\left( { - 681\mu^{2} + 204\mu - 36} \right) + \Delta \left[ {\mu_{0}^{2} - \left( {\frac{{1}}{2} - \frac{{1}}{4}\mu } \right)\mu_{0} } \right]. \\ \end{aligned} $$

We can easily find that $- 681\mu^{2} + 204\mu - 36 < 0$ and $\mu_{0}^{2} - \left( {\frac{{1}}{2} - \frac{{1}}{4}\mu } \right)\mu_{0} \le 0$ for any $\mu_{1} < \mu \le \frac{{2}}{3}, 0 < \mu_{0} < \frac{{1}}{2} - \frac{{1}}{4}\mu$, so we have $SW^{D} < 0$.

• (3) When $\frac{{2}}{3} < \mu \le 1, 0 < \mu_{0} < \frac{{2}}{5} - \frac{{1}}{5}\mu$, the difference in industry profits with and without transparency is given by

$$ \begin{aligned} & \pi^{D} = \pi^{with} - \pi^{without} \\ & \quad = \frac{{12\left( {c - 2\Delta } \right)^{2} + 25c_{0}^{2} }}{50\Delta } + \frac{{2\left( {2c + \Delta } \right)^{2} }}{25\Delta } - \frac{{12\left( {c - 2\Delta } \right)^{2} }}{49\Delta } - \frac{{2\left( {3c + \Delta } \right)^{2} }}{49\Delta } \\ & \quad = \frac{{8\left( { - 16c^{2} + 29c\Delta + 6\Delta^{2} } \right) + 1225c_{0}^{2} }}{2450\Delta } > 0. \\ \end{aligned} $$

The consumer surplus difference is

$$ \begin{aligned} & CS^{D} = CS^{with} - CS^{without} \\ & \quad = \frac{{1225c_{0}^{2} + 490c_{0} \left( {c - 2\Delta } \right) - 2\left( {c - 2\Delta } \right)\left( {c - 72\Delta } \right)}}{2450\Delta } \\ & \quad = \frac{\Delta }{2450}\left[ {1225\mu_{0}^{2} + \left[ {490\mu_{0} - 2\mu + 144\Delta } \right]\left( {\mu - 2} \right)} \right] < 0. \\ \end{aligned} $$

The social welfare difference is

$$ \begin{aligned} & SW^{D} = \pi^{D} + CS^{D} \\ & \quad = \frac{{ - 13c^{2} + 38c\Delta - 24\Delta^{2} + 49cc_{0} - 98\Delta c_{0} + 245c_{0}^{2} }}{245\Delta } \\ & \quad = \frac{\Delta }{245}\left( { - 13\mu^{2} + 38\mu - 24} \right) + \Delta \left[ {\mu_{0}^{2} - \left( {\frac{{2}}{5} - \frac{{1}}{5}\mu } \right)\mu_{0} } \right]. \\ \end{aligned} $$

We can easily determine that $\mu_{0}^{2} - \left( {\frac{{1}}{2} - \frac{{1}}{4}\mu } \right)\mu_{0} \le 0$ for any $\mu_{1} < \mu \le \frac{{2}}{3}, 0 < \mu_{0} < \frac{{1}}{2} - \frac{{1}}{4}\mu$. When $\mu \le \frac{12}{13}$, $- 13\mu^{2} + 38\mu - 24 \le 0$; and when $\mu > \frac{12}{13}$, $- 13\mu^{2} + 38\mu - 24 > 0$. So we have $SW^{D} < 0$ when $\mu \le \frac{12}{13}$. Thus we obtain the announced Proposition 8.

Proof of Proposition 9

(1) when $0 < \mu \le \frac{{1}}{3}, 0 < \mu_{0} < \frac{{1}}{5} + \frac{{1}}{5}\mu$, the difference in industry profits with and without transparency is given by

$$ \begin{aligned} & \pi^{D} = \pi^{with} - \pi^{without} \\ & \quad = \frac{{2\left( {2c - 3\Delta } \right)^{2} }}{25\Delta } + \frac{{12\left( {c + \Delta } \right)^{2} + 25c_{0}^{2} }}{50\Delta } - \frac{{2\left( {3c - 4\Delta } \right)^{2} }}{49\Delta } - \frac{{12\left( {c + \Delta } \right)^{2} }}{49\Delta } \\ & \quad = \frac{{ - 8\left( {16c - 19\Delta } \right)\left( {c + \Delta } \right) + 1225c_{0}^{2} }}{2450\Delta } > 0. \\ \end{aligned} $$

The consumer surplus difference is

$$ \begin{aligned} & CS^{D} = S^{with} - S^{without} \\ & \quad = \frac{{ - 2c^{2} - 144c\Delta - 142\Delta^{2} - 490cc_{0} - 490\Delta c_{0} + 1225c_{0}^{2} }}{2450\Delta } \\ & \quad = \frac{\Delta }{2450}\left( { - 2\mu^{2} - 144\mu - 142} \right) + \Delta \left[ {\frac{{1}}{2}\mu_{0}^{2} - \left( {\frac{{1}}{5} + \frac{{1}}{5}\mu } \right)\mu_{0} } \right]. \\ \end{aligned} $$

We can easily determine that $- 2\mu^{2} - 144\mu - 142 < 0$ and $\frac{{1}}{2}\mu_{0}^{2} - \left( {\frac{{1}}{5} + \frac{{1}}{5}\mu } \right)\mu_{0} \le 0 \;\;{\text{when}}\;\; 0 < \mu \le \frac{{1}}{3}, 0 < \mu_{0} < \frac{{1}}{5} + \frac{{1}}{5}\mu$. Hence we have ${\text{C}}S^{D} < 0$.

The social welfare difference is

$$ \begin{aligned} & SW^{D} = \pi^{D} + S^{D} \\ & \quad = \frac{{ - 13c^{2} - 12c\Delta + \Delta^{2} - 49cc_{0} - 49\Delta c_{0} + 245c_{0}^{2} }}{245\Delta } \\ & \quad = \frac{\Delta }{245}\left( { - 13\mu^{2} - 12\mu + 1} \right) + \Delta \left[ {\mu_{0}^{2} - \left( {\frac{{1}}{5} + \frac{{1}}{5}\mu } \right)\mu_{0} } \right]. \\ \end{aligned} $$

It is easily to verify that $\mu_{0}^{2} - \left( {\frac{{1}}{5} + \frac{{1}}{5}\mu } \right)\mu_{0} \le 0$ for any $0 < \mu \le \frac{{1}}{3}, 0 < \mu_{0} < \frac{{1}}{5} + \frac{{1}}{5}\mu$. When $\frac{{1}}{13} \le \mu \le \frac{{1}}{3}$, $- 13\mu^{2} - 12\mu + 1 \le 0$; and when $\mu < \frac{{1}}{13}$, $- 13\mu^{2} - 12\mu + 1 > 0$. So we have ${\text{S}}W^{D} < 0$ when $\frac{{1}}{13} \le \mu \le \frac{{1}}{3}$.

• (2) When $\frac{{1}}{3} < \mu \le \mu_{2} , 0 < \mu_{0} < \frac{{1}}{4} + \frac{{1}}{4}\mu$, the difference in industry profits with and without transparency is expressed by

$$ \begin{aligned} & \pi^{D} = \pi^{with} - \pi^{without} \\ & \quad = \frac{{\left( {c - 3\Delta } \right)^{2} }}{16\Delta } + \frac{{3c^{2} + 30c\Delta + 11\Delta^{2} + 16c_{0}^{2} }}{32\Delta } - \frac{{2\left( {3c - 4\Delta } \right)^{2} }}{49\Delta } - \frac{{12\left( {c + \Delta } \right)^{2} }}{49\Delta } \\ & \quad = \frac{{ - 715c^{2} + 1650c\Delta + 13\Delta^{2} + 784c_{0}^{2} }}{1568\Delta } > 0. \\ \end{aligned} $$

The consumer surplus difference is

$$ \begin{aligned} & CS^{D} = S^{with} - S^{without} \\ & \quad = \frac{{392c_{0}^{2} - 196c_{0} \left( {c + \Delta } \right) + \left( {c + \Delta } \right)\left( {17c - 263\Delta } \right)}}{784\Delta } \\ & \quad = \frac{\Delta }{784}\left( {17\mu^{2} - 246\mu - 263} \right) + \Delta \left[ {\frac{{1}}{2}\mu_{0}^{2} - \left( {\frac{{1}}{4} + \frac{{1}}{4}\mu } \right)\mu_{0} } \right]. \\ \end{aligned} $$

We can easily determine that $17\mu^{2} - 246\mu - 263 < 0$ and $\frac{{1}}{2}\mu_{0}^{2} - \left( {\frac{{1}}{4} + \frac{{1}}{4}\mu } \right)\mu_{0} \le 0 {\text{when }}\frac{{1}}{3} < \mu \le \mu_{2} , 0 < \mu_{0} < \frac{{1}}{4} + \frac{{1}}{4}\mu$. So we have $CS^{D} < 0$.

The social welfare difference is

$$ \begin{aligned} & SW^{D} = \pi^{D} + S^{D} \\ & \quad = \frac{{ - 681c^{2} + 1158c\Delta - 513\Delta^{2} - 392\left( {c + \Delta } \right)c_{0} + 1568c_{0}^{2} }}{1568\Delta } \\ & \quad = \frac{\Delta }{1568}\left( { - 681\mu^{2} + 1158\mu - 513} \right) + \Delta \left[ {\mu_{0}^{2} - \left( {\frac{{1}}{4} + \frac{{1}}{4}\mu } \right)\mu_{0} } \right]. \\ \end{aligned} $$

We can easily determine that $- 681\mu^{2} + 1158\mu - 513 < 0$ and $\mu_{0}^{2} - \left( {\frac{{1}}{4} + \frac{{1}}{4}\mu } \right)\mu_{0} \le 0 {\text{when }}\frac{{1}}{3} < \mu \le \mu_{2} , 0 < \mu_{0} < \frac{{1}}{4} + \frac{{1}}{4}\mu$. So we have $SW^{D} < 0$.

• (3) When $\mu_{2} < \mu \le 1, 0 < \mu_{0} < \frac{{1}}{4} + \frac{{1}}{4}\mu$, the difference in industry profits with and without transparency is given by

$$ \begin{aligned} & \pi^{D} = \pi^{with} - \pi^{without} \\ & \quad = \frac{{\left( {c - 3\Delta } \right)^{2} }}{16\Delta } + \frac{{3c^{2} + 30c\Delta + 11\Delta^{2} + 16c_{0}^{2} }}{32\Delta } - \frac{{\left( {c - 2\Delta } \right)^{2} }}{9\Delta } - \frac{{1}}{18}\left( {20c + \frac{{4c^{2} }}{\Delta } + 7\Delta } \right) \\ & \quad = \frac{{ - 17c^{2} - 10c\Delta + 7\Delta^{2} + 48c_{0}^{2} }}{96\Delta } < 0. \\ \end{aligned} $$

The consumer surplus difference is

$$ \begin{aligned} & S^{D} = S^{with} - S^{without} \\ & \quad = \frac{{c^{2} - 22c\Delta - 23\Delta^{2} - 36cc_{0} - 36\Delta c_{0} + 72c_{0}^{2} }}{144\Delta } \\ & \quad = \frac{\Delta }{144}\left( {\mu^{2} - 22\mu - 23} \right) + \Delta \left[ {\frac{{1}}{2}\mu_{0}^{2} - \left( {\frac{{1}}{4} + \frac{{1}}{4}\mu } \right)\mu_{0} } \right]. \\ \end{aligned} $$

It is easily determined that $\mu^{2} - 22\mu - 23 < 0$ and $\frac{{1}}{2}\mu_{0}^{2} - \left( {\frac{{1}}{4} + \frac{{1}}{4}\mu } \right)\mu_{0} \le 0 {\text{when }}\mu_{2} < \mu \le 1, 0 < \mu_{0} < \frac{{1}}{4} + \frac{{1}}{4}\mu$. So we have $CS^{D} < 0$.

The social welfare difference is

$$ \begin{aligned} & SW^{D} = \pi^{D} + S^{D} \\ & \quad = \frac{{ - 719c^{2} + 746c\Delta - 299\Delta^{2} - 294\left( {c + \Delta } \right)c_{0} + 1764c_{0}^{2} }}{1176\Delta } \\ & \quad = \frac{\Delta }{1176}\left( { - 719\mu^{2} + 746\mu - 299} \right) + \Delta \left[ {\mu_{0}^{2} - \left( {\frac{{1}}{4} + \frac{{1}}{4}\mu } \right)\mu_{0} } \right]. \\ \end{aligned} $$

It is easily detemined that $- 719\mu^{2} + 746\mu - 299 < 0$ and $\mu_{0}^{2} - \left( {\frac{{1}}{4} + \frac{{1}}{4}\mu } \right)\mu_{0} \le 0 {\text{when }}\mu_{2} < \mu \le 1, 0 < \mu_{0} < \frac{{1}}{4} + \frac{{1}}{4}\mu$. So we have $SW^{D} < 0$. Thus we arrive at the announced Proposition 9.