Switching Costs in Two-sided Markets - Ecares

This equation shows that the number of agents from side i belonging to platform 0 in thesecond period is the sum of three terms. The first element represents with probability µ ithe old customers are loyal. The second and third terms represent with probability 1−µ ithe old customers are switchers (whose preferences are unrelated in the two periods):they can either be switchers who did not switch away from platform 0 or switchers whoactually switched from platform 1 to platform 0.Then solving for n A 0,2 and nB 0,2 simultaneously, we obtain the second-period marketshares of groups A and B as follows:wheren i 0,2 = γ + β i + (1 − µ i )(p i 1,2 − pi 0,2 ) + e i(1 − µ i )(1 − µ j )(p j 1,2 − pj 0,2 ),2γγ =1 − (1 − µ A )(1 − µ B )e A e B ,β i =(2n i 0,1 − 1)(µ i + (1 − µ i )s i ) + (2n j 0,1 − 1)(1 − µ i)e i (µ j + (1 − µ j )s j ).Since I have assumed at the beginning that e i < 1, so γ > 0.The respective second-period profit of platforms 0 and 1 can be written as:π 0,2 = p A 0,2n A 0,2 + p B 0,2n B 0,2,π 1,2 = p A 1,2(1 − n A 0,2) + p B 1,2(1 − n B 0,2).Substitute n A 0,2 (pA 0,2 , pA 1,2 , pB 0,2 , pB 1,2 ) and nB 0,2 (pA 0,2 , pA 1,2 , pB 0,2 , pB 1,2 ) into these two profit functions.Then, by differentiating π 0,2 with respect to p A 0,2 and pB 0,2 , and differentiating π 1,2with respect to p A 1,2 and pB 1,2 , we obtain 4 equations and 4 unknowns.∂π 0,2∂p i 0,2∂π 1,2∂p i 1,2= n i 0,2 − pi 0,22γ (1 − µ i) − pj 0,22γ e j(1 − µ i )(1 − µ j ),= 1 − n i 0,2 − pi 1,22γ (1 − µ i) − pj 1,22γ e j(1 − µ i )(1 − µ j ).Solving the system of first-order conditions, one finds the following four second-periodequilibrium prices.wherep i 0,2 = 1 − e j(1 − µ i )1 − µ i+ η iλ i + ɛ i λ j(1 − µ i )∆ , (1)p i 1,2 = 1 − e j(1 − µ i )1 − µ i− η iλ i + ɛ i λ j(1 − µ i )∆ . (2)∆ = 9 − (1 − µ A )(1 − µ B )(e A + 2e B )(e B + 2e A ) > 0,λ i = (2n i 0,1 − 1)(µ i + (1 − µ i )s i ),η i = 3 − e j (e j + 2e i )(1 − µ i )(1 − µ j ) > 0,ɛ i = (1 − µ i )(e i − e j ).6