Procompetitive dual pricing

Abstract

We analyze the procompetitive effects of dual pricing, that is, input market price discrimination. An upstream firm has an incentive to maintain competition downstream which is realized by selling at an advantageous price to an inefficient downstream firm when discrimination is possible while it would exit under uniform pricing. We augment the exit issue into existing frameworks of Katz (Am Econ Rev 77:154–167, 1987), DeGraba (Am Econ Rev 80:1246–1253, 1990), and Yoshida (Am Econ Rev 90:240–246, 2000) which allows us to show that price discrimination in intermediary goods markets tends to have positive effects on allocative, dynamic and productive efficiency, respectively. In contrast, a discrimination ban tends to facilitate exit of relatively inefficient firms, thereby strengthening downstream market concentration.

Appendix

In this Appendix we present the omitted proofs and we derive Assumptions 2 and 3.

Proof of Lemma 2

We have to compare the upstream monopolist’s profit depending on whether or not the less efficient downstream firm 2 is served. With the monopoly input price given by (5) firm 2 remains active for all $\Delta <2(a-c)/7$, in which case the upstream monopoly profit becomes $L_{NE}^{U}=\left[ 2(a-c)-\Delta \right] ^{2}/24$. Note that the upstream monopolist’s profit is strictly decreasing in $\Delta$.

If, however, the monopolist prefers to serve only the efficient downstream firm, then its profit maximizing input price is $w_{E}^{U}=(a-c)/2$, for all $\Delta \ge (a-c)/4$. However, for all $\Delta <(a-c)/4$ the inefficient firm would purchase inputs at a price of $w_{E}^{U}$. In those cases, therefore, the input monopolist has to charge the exit inducing input price $\overline{w}$ if he wants to ensure that only one firm is served. Clearly, the monopoly profit at $\overline{w}$ is strictly smaller than the profit at $w_{E}^{U}$, which is given by $L_{E}^{U}(w_{E}^{U})=\left( a-c\right) ^{2}/8$. Note that this expression is independent of $\Delta$. Comparing $L_{NE}^{U}$ and $L_{E}^{U}$, we obtain the unique threshold value $\widetilde{\Delta } =(2-\sqrt{3})(a-c)$, with $L_{NE}^{U}<L_{E}^{U}$, for all $\Delta >\widetilde{\Delta }$, and $L_{NE}^{U}>L_{E}^{U}$, for all $\Delta <\widetilde{\Delta }$. Note that $\widetilde{\Delta }>(a-c)/4$, so that $w_{E}^{U}$ is a feasible pricing option for the monopolist for all $\Delta >\widetilde{\Delta }$. In addition, $\widetilde{\Delta }<2(a-c)/7$. Hence, for all $\Delta \in [\widetilde{\Delta },2(a-c)/7]$ the input monopolist decides to serve only firm 1 at $w_{E}^{U}$ even though he could also serve both firms at $w_{NE}^{U}$. It follows that the monopolist sets the exit inducing input price, $w_{E}^{U}$, for all $\Delta \ge \widetilde{\Delta }$, and the monopoly input price, $w_{NE}^{U}$, with both firms being active for all $\Delta <\widetilde{\Delta }$. $\square$

Proof of Proposition 1

Part (i): We have to compare social welfare, $W_{\psi }^{R}$, (the sum of upstream and downstream producer surplus plus consumer surplus) under the two regimes $R\in \{D,U\}$. Under price discrimination, D, social welfare is defined as $W^{D} =L^{D}+\sum _{i}\pi _{i}^{D}+CS^{D}$, which gives $W^{D}=(20(a-c)^{2} -20\Delta (a-c)+23\Delta ^{2})/72$. Similarly, with one firm active under uniform input pricing welfare is given by $W_{E}^{U}=L_{E}^{U}+\pi _{1,E} ^{U}+CS_{E}^{U}=7\left( a-c\right) ^{2}/32$. Solving $W^{D}-W_{E}^{U}=0$ we obtain the threshold value $\Delta ^{U}:=17(a-c)/46$. Moreover, welfare under regime U with both firms active can be written as $W_{NE}^{U}=(20(a-c)^{2} -20\Delta (a-c)+41\Delta ^{2})/72$. The welfare comparison then yields $W^{D}-W_{NE}^{U}<0$. Hence, $W_{NE}^{U}>W^{D}$ holds for all $\Delta \in (0,\widetilde{\Delta })$.

Part (ii): Equilibrium consumer surplus, $CS^{R}$, under the two regimes, $R\in \{D,U\}$, is increasing with total output, $Q^{R}$, with $CS^{R}=(Q^{R})^{2}/2$. Comparison yields that total output (and hence, consumer surplus) is always at least as large under regime D as under regime U, as $Q^{D}=Q_{E}^{U}=\left[ 3(a-c)-\Delta \right] /8$ is strictly larger than $Q_{NE}^{U}=(a-c)/3$ for all $\Delta <\widehat{\Delta }$. It follows that $CS^{D}>CS_{NE}^{U}$ if $\Delta >\widetilde{\Delta }$ and $CS^{D}=CS_{E}^{U}$ if $\Delta \le \widetilde{\Delta }$.

Proof of Proposition 2

The proof proceeds in two steps. Part (i) shows that $w^{U}=(a+7v)/6$ is the unique equilibrium and part (ii) proves the orderings of consumer surplus and social welfare.

Part (i): For the exit-inducing input price $w^{U}=(a+7v)/6$ the supplier realizes the profit level $L^{U}=\frac{1}{72}(a+7v)(5a-7v)$. Firstly, by not matching firm 1’s outside option, the supplier could gain at most $L^{OO}$. Since $L^{U}-L^{OO}=(5a^{2}+28av-49v^{2}-3a-3v)/72>0$, the upstream monopolist wants to prevent firm 1 from picking up its outside option. Secondly, the supplier wants to set a uniform input price which induces exit due to the following reasoning. To see this, suppose that firm 1 has no buyer power (i.e., the outside option never binds). Firm 1’s profit is then given by $\pi _{1}=(a-2w+\alpha w)^{2}/9$ and thus increases in the input price w. The supplier’s profit, however, is given by $L(w)=(aw-2w^{2}+2\alpha w^{2}+a\alpha w-2\alpha ^{2}w^{2})/3$ which has a strict maximum at $w^{NOO}=a(1+\alpha )/(4-4\alpha +4\alpha ^{2})$, while $\partial L/\partial w<0$, for $w>w^{NOO}$.

Next, let $\widehat{w}=(a-7v)/(4\alpha -8)$ denote the input price for which both firms are active and for which firm 1’s outside option is binding (i.e., $\widehat{w}:\pi _{1}(q_{1}(w,w),q_{2}(w,w)=\pi _{1}^{OO}$). Note that $v<4a/91$ and $\alpha >\alpha ^{\prime }$ ensure $w^{NOO}<\widehat{w}$. Consequently, if firm 1 has buyer power and the upstream monopolist wants to supply both firms, it will set $w=\widehat{w}$ since firm 1 would not accept a lower uniform input price and $L(w)<L(\widehat{w})$ for $w>\widehat{w}$. The input price $\widehat{w}$ ensures the profit $L(\widehat{w})=\frac{1}{24}(a-7v)(-5a-\alpha a+7v-7v\alpha +\alpha ^{2}a+7v\alpha ^{2})/(\alpha -2)^{2}$. The difference

$$\begin{aligned} L(\widehat{w})-L^{U} = \frac{(-7a+2\alpha a-7v+14v\alpha )(\alpha a-5a+7v+7v\alpha )}{72(\alpha -2)^{2}} \end{aligned}$$

is negative for all $\alpha <\alpha ^{\prime \prime }$. Thus, the supplier strictly prefers to offer $w^{U}$, so that exit occurs.

Part (ii): We compare the equilibrium values of consumer surplus and welfare under the different regimes. For regime $R\in \{D,U\}$ and market structure $\psi =\{E,NE\}$ we denote the overall produced equilibrium quantity $Q_{\psi }^{R}$. For the parameter ranges under consideration, regime D prevents exit and we obtain

$$\begin{aligned} CS_{NE}^{D}=\frac{(Q_{NE}^{D})^{2}}{2}=\frac{1}{2}\left( \frac{11}{24} a-\frac{7}{24}\right) ^{2}{\text { and }}\,CS_{E}^{U}=\frac{(Q_{E}^{R})^{2}}{2}=\frac{1}{2}\left( \frac{a+7v}{6}\right) ^{2} \end{aligned}$$

Concerning welfare we obtain

$$\begin{aligned} W_{NE}^{D}=Q_{NE}^{D}\left( a-\frac{Q_{NE}^{D}}{2}\right) =\frac{(11a-7v)(37a+7v)}{1152}. \end{aligned}$$

and

$$\begin{aligned} W_{E}^{U}=Q_{E}^{R}\left( a-\frac{Q_{E}^{R}}{2}\right) =\frac{(19a+7v)(5a-7v)}{288}. \end{aligned}$$

We find that $CD_{NE}^{D}-CS_{E}^{U}=\frac{35}{384}a^{2}-\frac{21}{64} av-\frac{245}{384}v^{2}>0$ holds for all $v<\frac{a}{5}$ and $W_{NE}^{D} -W_{E}^{U}=\frac{3}{128}a^{2}+\frac{35}{192}av+\frac{49}{384}v^{2}$ holds for all $v>0$, which means that the discriminatory pricing regime is beneficial both for consumers and for overall welfare. $\square$

Proof of Proposition 3

It is straightforward to calculate firm 1’s innovation incentives $\Psi ^{R}(\psi _{\theta },\psi _{0})=\pi _{1} ^{R}(\psi _{\theta })-\pi _{1}^{R}(\psi _{0})$, where the argument $\psi _{\theta }$ stands for the downstream market structure after the innovation and $\psi _{\theta }$ indicates the market structure before the innovation, with $\psi _{(.)}\in \{NE,E\}$. We then obtain

$$\begin{aligned} \Psi ^{D}(NE,NE)&=\frac{1}{36}\left[ (a-c+\Delta +2\theta )^{2}-(a-c+\Delta )^{2}\right] ,\quad {\text {for }}\,\Delta <\Delta ^{\prime \prime }\text {, and }\\ \Psi ^{D}(E,NE)&=\frac{1}{16}(a-c+\theta )^{2}-\frac{1}{36}(a-c+\Delta )^{2},\quad {\text {for }}\,\widehat{\Delta }>\Delta \ge \Delta ^{\prime \prime }.\end{aligned}$$

Similarly, we can calculate firm 1’s innovation incentives $\Psi ^{U} (\psi _{\theta },\psi _{0})$ under regime U, which are given by

$$\begin{aligned} \Psi ^{U}(NE,NE)&=\frac{1}{144}\left[ (2(a-c)+5\Delta +7\theta )^{2} -(2(a-c)+5\Delta )^{2}\right] ,\quad{\text {for }}\,\Delta <\Delta ^{\prime },\\ \Psi ^{U}(E,NE)&=\frac{1}{16}(a-c+\theta )^{2}-\frac{1}{144}(2(a-c)+5\Delta )^{2},\quad{\text {for }}\,\widetilde{\Delta }>\Delta \ge \Delta ^{\prime },\\ \Psi ^{U}(E,E)&=\frac{1}{16}\left[ (a-c+\theta )^{2}-(a-c)^{2}\right] ,\quad{\text {for }}\,\widehat{\Delta }>\Delta \ge \widetilde{\Delta }. \end{aligned}$$

We now have to pairwise compare $\Psi ^{U}(\psi _{\theta },\psi _{0})$ and $\Psi ^{D}(\psi _{\theta },\psi _{0})$.

Parts (i) and (ii): Given that $\Delta ^{\prime \prime }>\Delta ^{\prime }$ for all $\theta >0$ we have to compare $\Psi ^{U}(NE,NE)$ versus $\Psi ^{D}(NE,NE)$ for the case where $\Delta <\Delta ^{\prime }$. In addition, we have to compare $\Psi ^{U}(E,E)$ versus $\Psi ^{D}(NE,NE)$ and $\Psi ^{D}(E,NE)$ for the cases where $\Delta \ge \widetilde{\Delta }$. Firstly, note that $\Psi ^{U}(NE,NE)-\Psi ^{D}(NE,NE)>0$ can be rewritten as $4\theta (a-c)+18\Delta \theta +11\theta ^{2}>0$, which is clearly always fulfilled. Secondly, we can rewrite $\Psi ^{U}(E,E)-\Psi ^{D}(E,NE)<0$ as $\frac{1}{144}(5a-5c+2\Delta )(-a+c+2\Delta )<0$. This inequality unambiguously holds for all $\Delta <\widehat{\Delta }$ which we have assumed in A1. And thirdly, note that $\Psi ^{U}(E,E)-\Psi ^{D}(NE,NE)<0$ if $\theta (-7\theta +2a-2c-16\Delta )/144<0$ which holds for $\Delta \ge \widetilde{\Delta }$.

Part (iii): We can rewrite $\Psi ^{D}(E,NE)-\Psi ^{U}(E,NE)>0$ as $\frac{1}{48}\Delta (4a-4c+7\Delta )>0$, which always holds.

We show that $\Psi ^{D}(NE,NE)-\Psi ^{U}(E,NE)>0$ if and only if $\Delta \in [\Delta ^{*},\min \{ \Delta ^{\prime \prime },\widetilde{\Delta }\})$. Firstly, let us derive $\Delta ^{*}$. The equation $\Psi ^{D}(NE,NE)-\Psi ^{U}(E,NE)=0$ is equivalent to $-5(a-c)^{2}/144-\theta (a-c)/72+5\Delta (a-c)/36+(5\Delta /12+2\theta /15)^{2}+37\theta ^{2}/1200=0$. Note that the left-hand side of this equation is increasing in $\Delta$ and that

$$\begin{aligned} \Delta ^{\prime \prime \prime }=-\frac{2}{5}(a-c)-\frac{8}{25}\theta +\frac{1}{25}\sqrt{-111\theta ^{2}+210\theta (a-c)+(15a-15c)^{2}} \end{aligned}$$

is the only feasible non-negative solution for the equation. Secondly, note that $d\Delta ^{\prime \prime \prime }/d\theta <0,d\widetilde{\Delta }/d\theta =0$ and $\Delta ^{\prime \prime \prime }(\theta )<\Delta ^{\prime \prime }(\theta )$ as well as $d\Delta ^{\prime }/d\theta <0$ for all $\theta$. Thirdly, note that at $\theta =0$ we obtain $\Delta ^{\prime }(0)=\widetilde{\Delta }>\Delta ^{\prime \prime \prime }(0)$. Fourthly, there exists a unique $\widehat{\widehat{\theta }}\in (0,\widehat{\theta })$ with $\Delta ^{\prime \prime \prime }(\widehat{\widehat{\theta }})=\Delta ^{\prime }(\widehat{\widehat{\theta }})$, which is $\widehat{\widehat{\theta }}=(a-c)(171\sqrt{3}-4-\sqrt{35069+21532\sqrt{3}}))/129\approx 0.101(a-c)$. Thus, $\Delta ^{\prime \prime \prime }(\theta )<\Delta ^{\prime }(\theta )$ for $\theta <\widehat{\widehat{\theta }}$ and $\Delta ^{\prime \prime \prime }(\theta )\in [\Delta ^{\prime }(\theta ),\min \{ \Delta ^{\prime \prime },\widetilde{\Delta }\})$ for $\theta >\widehat{\widehat{\theta }}$. We define $\Delta ^{*}=\max \{ \Delta ^{\prime \prime \prime }(\theta ),\Delta ^{\prime }(\theta )\}$.

Proof of Proposition 4

According to Proposition 3, it is possible that an innovation project $I(\theta )$ is exclusively carried out under D, but not under regime U. Given $\Delta \in [\Delta ^{*},\widetilde{\Delta })$, then it follows from Lemma 2 that firm 2 stays in the market if the innovation is not carried out under regime U. In that case, total output is given by $Q_{NE}^{U}=\left[ 2(a-c)-\Delta \right] /6$. If the innovation is undertaken under regime D, equilibrium quantities can be computed as $Q_{NE}^{D}=(2(a-c)+\theta -\Delta )/6$ for $\Delta <\Delta ^{\prime \prime }$ and $Q_{E}^{D}=(a-c+\theta )/4$ for $\Delta >\Delta ^{\prime \prime }$. Obviously, $Q_{NE}^{D}>Q_{NE}^{U}$ holds always for $\theta >0$. Note that $Q_{E}^{D}>Q_{NE}^{U}$ for $\Delta >\Delta ^{\prime \prime }$ since $\Delta ^{\prime \prime }>(a-c)/12$. Thus, consumer surplus is higher under the discriminatory regime.

Given $\Delta \in [\widetilde{\Delta },\widehat{\Delta })$, according to Proposition 3, it may also happen that the innovation is exclusively taken out under regime D. Under both regimes only firm 1 is active and its produced quantity is strictly increasing in $\theta$. Thus, consumer surplus is higher under regime D.

Also social welfare can be higher under the discriminatory regime if the innovation is large enough. For instance $a=1$, $c=0.05$, $\theta =0.05$ gives that welfare is strictly higher under the discriminatory regime for all parameter values $\Delta$. $\square$