Price-Increasing Competition on Two-Sided Markets with Homogeneous Platforms

Abstract

We make a case for price-increasing competition on “competitive bottleneck” two-sided markets. We argue that demand interrelation might be sufficient to cause either no observable price effect of competition or price-increasing competition. Under price equality, total demand on both market sides in the duopoly market exceeds total demand in the monopoly market. Furthermore, even though there is no observable price effect, there is still a competitive effect that becomes manifest in total duopoly equilibrium profits being strictly smaller than monopoly profits. The relationship of total welfare is ambiguous in subsidization cases, while without subsidization, welfare is strictly greater in duopoly.

Appendices

Appendix 1

1.1 Quantities at the lower bound of the Nash-equilibria

Proposition 1 describes upper and lower bounds for equilibrium prices. Our exposition focused on the upper bound that yields positive equilibrium profits. In this appendix, we show that our qualitative results can also be obtained using the lowest equilibrium price that is when using the equilibrium

$$ \left( {p_a^{*} = p_a^M = \frac{{\overline \mu + \alpha }}{2},p_c^{*} = \underline p_c^{*} = \frac{{\left( {{{\overline \mu}^2} - {\alpha^2}} \right)}}{{{\eta^2} - 8}}} \right). $$

In this case, each platform realizes an advertising quantity of

$$ \underline n_a^{*} = \underline n_a^{{i*}} = \underline n_a^{{j*}} = {n_a}\left( {\underline p_c^{*},p_a^{*}} \right) = \frac{{2 \cdot \overline \theta \cdot \eta }}{{{\eta^2} - 8}},\;{\mathrm{\text and}}\;{\mathrm{\text attracts}}\;\underline n_c^{*} = \underline n_c^{{i*}} = \underline n_c^{{j*}} = {n_c}\left( {\underline p_c^{*},p_a^{*}} \right) = \frac{{8 \cdot \overline \theta }}{{8 - {\eta^2}}} $$

consumers. Respecting that $\underline \varPi_i^s, \underline n_c^s, \underline n_a^s,n_c^M,n_a^M,{\varPi^M}\mathop{ \geqslant}\limits^{!} 0$, $\underline p_c^{*} = p_c^M$ results for

$$ \matrix{ {\left( {\overline \mu, \alpha } \right) \in \left\{ {\left( {2\sqrt {2}, \sqrt {2} } \right),\left( {\sigma, \alpha > \sqrt {2} } \right)} \right\},\;{\mathrm{\text where}}} \\ {\sigma = {R_1}\,{\mathrm{of}}\,e \cdot {x^3} + \left( {6 - 3 \cdot {e^2}} \right) \cdot {x^2} + e \cdot \left( {3 \cdot {e^2} - 12} \right) \cdot x - {e^4} + 6 \cdot {e^2} - 16.} \\ }<!end array> $$

Comparing with (29), we see that $\alpha = \sqrt {2}$ is the lower bound of α, and that in this case $\max \left( {\beta, \delta } \right) = \delta = 2\sqrt {2}$, so that $\underline p_c^{*} = \overline p_c^{*}$, i.e. the equilibrium is unique for $\left( {\overline \mu, \alpha } \right) = \left( {2\sqrt {2}, \sqrt {2} } \right)$.

Comparison of monopoly and duopoly quantities reveals that $\underline n_c^{*} > n_c^M$, $\underline n_a^{*} < n_a^M < 2 \cdot \underline n_a^{*}$, and $2 \cdot {\underline \varPi^{*}} = 0 < {\varPi^M}$, which is consistent with our findings for the equilibrium $\left( {p_a^{*},\bar{p}_c^{*}} \right)$.

Price-increasing competition $\left( {\underline p_c^{*} > p_c^M} \right)$ results for

$$ \left( {\bar{\mu },\alpha } \right) \in \left\{ {\left( {\sigma < \bar{\mu } \leqslant \frac{{{\alpha^2} - 1 + \sqrt {{4 \cdot {\alpha^2} + 1}} }}{\alpha },\alpha > \sqrt {2} } \right)} \right\}. $$

As with $\left( {p_a^{*},\bar{p}_c^{*}} \right)$, $2 \cdot {\underline \varPi^{*}} = 0 < {\varPi^M}$ and $\underline n_c^{*} \leqslant n_c^M$ holds, while the relationship of $\underline n_c^{*}$ and $n_c^M$ and $n_a^M$ and $2 \cdot \underline n_a^{*}$ is ambiguous.

Appendix 2

2.1 Explicit collusion or merger on the duopoly market

Assume, both platform operators are able and willing to cooperate in order to maximize joint profits. Given (10) and (11), the operators have two options: Either they equally divide consumer demand between their platforms or they close down one platform and create a monopoly. In the first case -for reasons to be seen soon, we label it “hypothetical collusion case”- the optimization problem is

$$ \mathop{{\max }}\limits_{{{n_c},n_a^i}} \quad {\varPi_k} = \sum\limits_{{i = 1}}^2 {n_a^i \cdot \frac{{{n_c}}}{2} \cdot p_a^i\left( {n_a^i,\frac{{{n_c}}}{2}} \right) + \frac{{{n_c}}}{2} \cdot p_c^i\left( {n_a^i,{n_c}} \right)}, $$

which yields a maximum profit of

$$ {\varPi_k} = \frac{{2 \cdot {{\bar{\theta }}^2}}}{{8 - {\eta^2}}}, $$

optimal quantities

$$ n_c^k = - \frac{{4 \cdot \bar{\theta }}}{{{\eta^2} - 8}}\;{\mathrm{and}}\;n_a^{{1,k}} = n_a^{{2,k}} = n_a^k = \frac{{\eta \cdot \bar{\theta }}}{{{\eta^2} - 8}}, $$

and optimal prices

$$ p_c^{{1,k}} = p_c^{{2,k}} = p_c^{{i,k}} = \frac{{\left( {\eta \cdot \overline \mu + 4} \right) \cdot \overline \theta \bar{\theta }}}{{8 - {\eta^2}}}\;{\mathrm{and}}\;p_a^{{1,k}} = p_a^{{2,k}} = p_a^{{i,k}} = p_a^M. $$

Given the nonnegativity constraints on $\overline \mu$ and $\overline \theta$, and the parameter restrictions implied by the nonnegativity of ${\varPi_k}$ and ${\varPi_M}$, the maximum hypothetical collusion profit never exceeds the optimal monopoly profit (9), and consumer prices never exceed monopoly consumer prices. Furthermore, there is only one corner solution, in which both profits and consumer prices become equal. Therefore, explicit collusion or merger always implies that the operators close down one platform to play the monopoly solution, except, if $\left( {\overline \mu, \alpha } \right) = \left( {\alpha, \alpha > 0} \right)$, in which case the operators are indifferent between keeping both platforms open and closing down one.

Assume that for some exogenous reason it is not possible to close down one platform. In case of a merger, this might be due to obligations of a regulating authority. To study the welfare effects in this case, we compute hypothetical consumer surplus as

$$ \matrix{ {\int\limits_0^{{n_c^k}} {p_c^k\left( {{n_c},n_a^k} \right)} \;d{n_c} - n_c^k \cdot p_c^k = \frac{{8 \cdot {{\overline \theta}^2}}}{{{{\left( {{\eta^2} - 8} \right)}^2}}},\;{\mathrm{\text where}}} \\ {p_c^1 = p_c^2 = p_c^k = \overline \theta - \alpha \cdot {n_a} - {n_c}.} \\ }<!end array> $$

Hypothetical advertiser surplus is

$$ \matrix{ {2 \cdot \left( {\int\limits_0^{{n_a^k}} {p_a^k\left( {{n_a},n_c^k} \right)\;} d{n_a} - n_a^k \cdot p_a^k} \right) = \frac{{{\eta^2} \cdot \overline \theta }}{{2 \cdot \left( {8 - {\eta^2}} \right)}},\;{\mathrm{\text where}}} \\ {p_a^1 = p_a^2 = p_a^k = \bar{\mu } - \frac{{2 \cdot {n_a}}}{{{n_c}}}.} \\ }<!end array> $$

Comparing consumer, advertiser, and producer surplus of the hypothetical collusion case and the monopoly optimum, we find that each of these welfare components is at least as great in the monopoly case as it is in the hypothetical collusion case. Therefore total welfare in the hypothetical collusion case also never exceeds total welfare in the monopoly case.

Comparing welfare outcomes of hypothetical collusion and duopoly equilibrium case, we need to distinguish the cases known from Section 4 and obtain the results presented in Table 6.

Table 6 Welfare in the case of explicit collusion