# Price discrimination

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):

$$v_3(q_3)-P_3≥v_3(q_2)-P_2$$

$$v_3(q_3)-P_3≥v_3(q_1)-P_1$$

$$v_2(q_2)-P_2≥v_2(q_3)-P_3$$

$$v_2(q_2)-P_2≥v_2(q_1)-P_1$$

$$v_1(q_1)-P_1≥v_1(q_3)-P_3$$

$$v_1(q_1)-P_1 \geq v_1(q_2)-P_2$$

Where I suppose my profit function is:

$$\pi=P_3+P_2+P_1-c(q_3+q_2+q_1)$$

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:

$$P_1=v_1(q_1)$$

$$P_3=v_1(q_1)+v_3(q_3)-v_3(q_1)$$

Any help would be appreciated. If anything is unclear just let me know, since I had a hard time articulating the question.

• I guess this is pretty much a case of non linear pricing with 3 types. But the only examples seem to be with two types. – user12783 Apr 6 '17 at 19:40
• Don't you have individual rationality constraints as well? I.e., for instance $P_1 \leq v(q_1)$? – Oliv Apr 7 '17 at 6:29
• Yes, but I guess these are more or less implied by the IC constraints. Maybe just except v(q1)-P1>0 – user12783 Apr 7 '17 at 6:37
• That is right, you need $v_1(q_1) \geq P_1$ but all other IR constraints will be implied by the IC constraints. Have you tried to apply the same methodology as in the two-agents case and guess which constraints must be binding? – Oliv Apr 7 '17 at 6:56