Platform competition with common ownership

Abstract

This paper investigates the allocative effects of platform competition with common ownership in two-sided markets. We find that when both sides singlehome, common ownership increases the consumer surplus for the side treated as a loss leader, but decreases the consumer surplus for the other side. When one side multihomes, common ownership does not affect the consumer surplus for the multihoming side, but decreases the consumer surplus for the singlehoming side. When we introduce demand expansion into the model, captive users may still benefit from common ownership if they are treated as a loss leader. We show that common ownership does not necessarily hurt all platform users by extending the canonical Armstrong model of platform competition.

Appendices

Appendix A

In the model with demand expansion of Sect. 4, the following assumption is necessary to ensure that both platforms are active, and the markets for both sides are fully covered.

Assumption 3.

$4t_{1} t_{2} \left( {2 - \frac{1}{{1 + \theta t_{1} }}} \right) > \left( {1 + \theta t_{1} } \right)\left( {\alpha_{1} + \alpha_{2} } \right)^{2} + 4\theta t_{1} \alpha_{1} \alpha_{2}$, $r_{1} > f_{1} + \frac{{1 - \lambda + 4\theta t_{1} }}{{1 - \lambda + 2\theta t_{1} }} \cdot \frac{{t_{1} - \alpha_{1} }}{2} + \frac{{t_{1} \left( {1 + \theta \alpha_{1} } \right) - \alpha_{2} \left( {1 + \theta t_{1} } \right)}}{{1 - \lambda + 2\theta t_{1} }}$, and $r_{2} > f_{2} + \frac{{t_{2} - \theta \alpha_{2}^{2} }}{2} + \frac{{t_{2} - \theta \alpha_{1} \alpha_{2} }}{1 - \lambda } - \frac{{\theta \alpha_{2} }}{{1 - \lambda + 4\theta t_{1} }}\left[ {\frac{{\left( {3 + \lambda } \right)\alpha_{1} }}{1 - \lambda } + \alpha_{2} } \right] - \left[ {1 + 2\theta \left( {r_{1} - f_{1} } \right) + \theta \left( {\alpha_{1} - \alpha_{2} } \right)} \right]\left[ {\frac{{\alpha_{1} + \theta t_{1} \left( {\alpha_{1} - \alpha_{2} } \right)}}{{1 - \lambda + 4\theta t_{1} }} + \frac{{\alpha_{2} }}{2}} \right]$.

We can find that if there is no demand expansion (i.e., $\theta = 0$), then Assumption 3 collapses to Assumption 1. The first condition is associated with active platforms, and it implies $t_{1} t_{2} > \left( {1 + \theta t_{1} } \right)\alpha_{1} \alpha_{2}$. When the markets are fully covered, demand expansion is likely to downward adjust the minimum intrinsic benefit of joining a platform for group 2, since the cross-group external effect for group 2 will increase. However, the minimum intrinsic benefit for side 1 depends on platform differentiation $t_{1}$. To see this, we define $r_{1}^{*} \left( \theta \right) = f_{1} + \frac{{1 - \lambda + 4\theta t_{1} }}{{1 - \lambda + 2\theta t_{1} }} \cdot \frac{{t_{1} - \alpha_{1} }}{2} + \frac{{t_{1} \left( {1 + \theta \alpha_{1} } \right) - \alpha_{2} \left( {1 + \theta t_{1} } \right)}}{{1 - \lambda + 2\theta t_{1} }}$, which implies $\frac{{\partial r_{1}^{*} \left( \theta \right)}}{\partial \theta } = \frac{{t_{1} \left( {1 + \lambda } \right)\left( {\alpha_{2} - t_{1} } \right)}}{{\left( {1 - \lambda + 2\theta t_{1} } \right)^{2} }}$. When platform differentiation is larger than the cross-group effect for group 2 (i.e., $t_{1} > \alpha_{2}$), demand expansion can reduce the minimum intrinsic benefit of joining a platform for side 1. Without hinterlands, according to Proposition 1, we know that $t_{1} > \alpha_{2}$ implies platforms make profits from group 1. To attract potential users by demand expansion, platforms should decrease the price, and thus a lower requirement for group 1’s intrinsic benefit.

Appendix B

Lemma 1.

When the market is fully covered and demand expansion is possible, if the net surplus generated by group 1’s marginal user joining a platform is negative (i.e., $r_{1} - f_{1} + \frac{{\alpha_{1} - t_{1} }}{2} < 0$), platforms use the side of group 1 as a loss leader.

Proof.

Substituting $r_{1} = f_{1} - \frac{{\alpha_{1} - t_{1} }}{2}$ into the second condition of Assumption 3, we have $\frac{{\left( {1 + \theta t_{1} } \right)\left( {\alpha_{2} - t_{1} } \right)}}{{1 - \lambda + 2\theta t_{1} }} > 0$. Let $r_{1}^{*} = f_{1} - \frac{{\alpha_{1} - t_{1} }}{2}$, then the second condition of Assumption 3 can be written as $r_{1} - r_{1}^{*} + \frac{{\left( {1 + \theta t_{1} } \right)\left( {\alpha_{2} - t_{1} } \right)}}{{1 - \lambda + 2\theta t_{1} }} > 0$. To ensure that the second condition of Assumption 3 is satisfied, if $r_{1} < r_{1}^{*}$, then there must be $\alpha_{2} > t_{1}$. In addition, when $r_{1} < f_{1} - \frac{{\alpha_{1} - t_{1} }}{2}$, we have $t_{1} > 2r_{1} - 2f_{1} + \alpha_{1}$, then $\frac{{1 + \theta \left( {2r_{1} - 2f_{1} + \alpha_{1} } \right)}}{{1 + \theta t_{1} }} \cdot t_{1} < t_{1}$. Hence, if $\alpha_{2} > t_{1}$, we have $\alpha_{2} > t_{1} > \frac{{1 + \theta \left( {2r_{1} - 2f_{1} + \alpha_{1} } \right)}}{{1 + \theta t_{1} }} \cdot t_{1}$, which implies platforms take the side with captive users as a loss leader (see Proposition 5). ■

Appendix C

In the symmetric equilibrium, the consumer surplus of Hotelling users on side 1, the consumer surplus of Hotelling users on side 2, and the consumer surplus of captive users can be given by:

$$CS_{1}^{H} = 2\int_{0}^{1/2} {\left( {r_{1} + \alpha_{1} n_{2}^{H} - p_{1}^{H} - t_{1} x} \right)dx} ,$$

$$CS_{2}^{H} = 2\int_{0}^{1/2} {\left( {r_{2} + \alpha_{2} n_{1}^{H} - p_{2}^{H} - t_{2} x} \right)dx} ,$$

$$CS_{c}^{H} = 2\int_{0}^{{\theta \left( {{{\alpha_{1} } \mathord{\left/ {\vphantom {{\alpha_{1} } 2}} \right. \kern-0pt} 2} + r_{1} - p_{1}^{H} } \right)}} {\left( {r_{1} + \alpha_{1} n_{2}^{H} - p_{1}^{H} - \frac{x}{\theta }} \right)dx} .$$

The profits of two platforms can be described by:

$$\Pi^{H} = 2\left( {p_{1}^{H} - f_{1} } \right)n_{1}^{H} + 2\left( {p_{2}^{H} - f_{2} } \right)n_{2}^{H} .$$

Social welfare can be expressed as the sum of the consumer surplus of Hotelling users on side 1, the consumer surplus of Hotelling users on side 2, the consumer surplus of captive users, and the profits of two platforms:

$$W = CS_{1}^{H} + CS_{2}^{H} + CS_{c}^{H} + \Pi^{H} .$$

The first-order derivative of social welfare with respect to the degree of common ownership can be calculated by:

$$\frac{dW}{{d\lambda }} = \frac{{dCS_{1}^{H} }}{d\lambda } + \frac{{dCS_{2}^{H} }}{d\lambda } + \frac{{dCS_{c}^{H} }}{d\lambda } + \frac{{d\Pi^{H} }}{d\lambda }.$$

According to Eqs. (22) and (23), we have $\frac{{dn_{1}^{H} }}{d\lambda } = - \theta \frac{{dp_{1}^{H} }}{d\lambda }$ and $\frac{{dn_{2}^{H} }}{d\lambda } = 0$. Using $\frac{{dn_{2}^{H} }}{d\lambda } = 0$ and combining Eqs. (22) and (23), we obtain:

$$\frac{{dCS_{1}^{H} }}{d\lambda } = - \frac{{dp_{1}^{H} }}{d\lambda },$$

$$\frac{{dCS_{2}^{H} }}{d\lambda } = \alpha_{2} \frac{{dn_{1}^{H} }}{d\lambda } - \frac{{dp_{2}^{H} }}{d\lambda },$$

$$\frac{{dCS_{c}^{H} }}{d\lambda } = - 2\theta \left( {\frac{{\alpha_{1} }}{2} + r_{1} - p_{1}^{H} } \right)\frac{{dp_{1}^{H} }}{d\lambda },$$

$$\frac{{d\Pi^{H} }}{d\lambda } = \left[ {1 + 2\theta \left( {\frac{{\alpha_{1} }}{2} + r_{1} - p_{1}^{H} } \right)} \right]\frac{{dp_{1}^{H} }}{d\lambda } + \frac{{dp_{2}^{H} }}{d\lambda } + 2\left( {p_{1}^{H} - f_{1} } \right)\frac{{dn_{1}^{H} }}{d\lambda }.$$

The above equations imply:

$$\frac{dW}{{d\lambda }} = \alpha_{2} \frac{{dn_{1}^{H} }}{d\lambda } + 2\left( {p_{1}^{H} - f_{1} } \right)\frac{{dn_{1}^{H} }}{d\lambda } = - \frac{\theta }{{1 - \lambda + 4\theta t_{1} }}\left( {p_{1}^{H} - f_{1} } \right)\left[ {2\left( {p_{1}^{H} - f_{1} } \right) + \alpha_{2} } \right],$$

where we have applied the relations $\frac{{dn_{1}^{H} }}{d\lambda } = - \theta \frac{{dp_{1}^{H} }}{d\lambda }$ and $\frac{{dp_{1}^{H} }}{d\lambda } = \frac{{p_{1}^{H} - f_{1} }}{{1 - \lambda + 4\theta t_{1} }}$.

The calculation suggests that the net effect of common ownership on social welfare depends on $\alpha_{2} \frac{{dn_{1}^{H} }}{d\lambda }$ and $2\left( {p_{1}^{H} - f_{1} } \right)\frac{{dn_{1}^{H} }}{d\lambda }$, which represent the cross-group effect of additional captive users on the other side, and the profits of platforms made from additional captive users, respectively.