If I have 3 consumers who I want to sell package 1, 2 and 3 to respectively. Meaning consumer 1 I would like to sell package 1 to, consumer 2 I would like to sell package 2 to and so on. The packages consist of the price and quantity of a good (let's say bananas) (p,q). Consumer 3 has demand D3, consumer 2 D2, and consumer D1, where the reservation prices can be described by the following relationship $v_3>v_2>v_1$.
In order for me to price discriminate my profit function needs to comply with some incentive compatibility constraints (I found 6 in total):
$$v_1(q_1)-P_1 \geq v_1(q_2)-P_2$$
Where I suppose my profit function is:
where c is the constant marginal cost.
How do I deduce the optimal prices?
The problem is I've only seen problems with two types of consumers where the optimal prices were:
Any help would be appreciated. If anything is unclear just let me know, since I had a hard time articulating the question.