It took me about a week. Finally I get the flow right.
We start the separating equilibrium part with the two necessary conditions. To characterize the equilibrium, we proceed in three steps. First, we specify the belief. This is an unusual treatment, but it is critical to simply the later exposition. Because R’s reactions— half of the story—depend on this belief.
The second step is to find the best retail and direct selling quantities. Here we use backward induction argument. We also lay out the optimization problem for each player.
The third step is to specify the equilibrium wholesale prices. The driving force here comes from the incentive compatibility constraints. Because type-L enjoys two advantages—the information advantage over R and the selling cost advantage over type-H—he has strong incentive to mislead. Therefore, for the equilibrium to be separating, type-L must prefer revealing than mimicking.
Got to finish the writing tomorrow.
[Jasper, Canada, 10/2015]