Quality Competition and Market Segmentation in the Security Software Market

Setting u n+1 = 1, from Figure 2, we can find the market coverage of θ j , j = 1, 2, …, n, as x j = u j+1 – u j , which can be summed over j to obtain u i = 1 – G n j =i x j. Substituting this into (2) for i = 1 and noting that p 0 = θ 0 = 0, we find (A1) p G u G x j j n 1 11 1 1 1 = = −       =  θ θ We will now prove the lemma by induction. It is clear that (3) reduces to (A1) for i = 1. Let (3) hold for i = k, implying p G x x k k j j k n j j j k = −  1 1 1 We substitute this into (2) for i = k +1 to obtain () () () p G u p G x G x x G x G x x x x k k k k k k k j j k n k j j k n j j j k k k j j k n k j j k n k k j j k k j k + + +   G x x k j j k n j j j k θ θ 1 1 1 1 In other words, if (3) holds for i = k, then it also holds for i = k + 1. Since (3) holds for i = 1, the proof is now complete. #