Switching Costs in Two-sided Markets - Ecares

3.4 Switching Costs, One-sided Markets and RationalitySuppose that all consumers and the two platforms are rational such that δ i , δ F > 0 andα i = 0. e i = 0 since there is no two-sidedness. For simplicity, assume that consumersdraw new preferences in each period (µ i = 0). Equation (3) can be simplified top i 0,1 = 1 + δ i s i + 4 3 δ is 2 i − 2 3 δ F s i .Differentiating p i 0,1 with respect to s i, we obtainLemma 1. In the two-period single-homing duopoly model, where network externality isabsent, all consumers are rational, and their preferences are independent over time,i. If consumers are relatively less patient than the platforms, then the relationshipbetween first-period equilibrium prices and switching costs is U-shaped.ii. If consumers are relatively more patient than the platforms, then first-period equilibriumprices always increase in switching costs.See Appendix B for the proof.3.5 Switching Costs, Two-sided Markets and RationalityThe preliminaries are the same as the previous case, δ i , δ F , s i > 0 and α i , µ i = 0, exceptthat the market is two-sided now (e i > 0). In this case, Equation (3) can be rewrittenasp i 2(3 − ∆)0,1 = 1 + δ i s i −∆δ is 2 i − 2δ j∆ (e j + 2e i )s j s i − δ F ζs i − e j .Differentiating p i 0,1 with respect to s i, we obtainLemma 2. In the two-period two-sided single-homing duopoly model, where all consumersare rational, both groups of consumers have switching costs, each side exerts thesame external benefit on the other side, and their preferences are independent over time,i. For sufficiently weak externality,• If consumers are relatively less patient than the platforms, then the relationshipbetween first-period equilibrium prices and switching costs is U-shaped.• If consumers are relatively more patient than the platforms, then first-periodequilibrium prices always increase in switching costs.ii. For sufficiently strong externality,• If consumers are relatively less patient than the platforms, then first-periodequilibrium prices always decrease in switching costs.13