Optimal Placement Strategy

Solving first-order conditions for the search engine's profit function, we see that the optimal degree of independence 35#35 is given by

 36#36 (3)

It can be verified that when 37#37, the profit function is concave and that the optimal 35#35 given by Eq.  satisfies second-order conditions for optimality. Since we require that 39#39, Eq.  gives the optimal solution when 40#40.

Lemma 1   For any function 14#14 this inequality yields an interval 41#41 such that for all 13#13 within this interval, Eq.  determines the optimal 35#35.

To further develop the results, we first state the following property.

Lemma 2   Let 42#42 represent the elasticity of 43#43 with respect to 13#13, hence 44#44. Then for every given 13#13 and 14#14, there exists a 45#45 such that if 46#46, then 47#47, and if 48#48, then 49#49. Specifically, this value 45#45 equals 50#50.

51#51
To understand this result, consider 42#42, the responsiveness of listing company's benefit to the users disutilities from paid placement strategy. When the responsiveness is relatively low, the search engine get limited increase in revenue due to the limited increase in benefit of the listing companies by hurting users utilities, so the search engine should be careful about his paid placement strategy.