<p>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 $v_3&gt;v_2&gt;v_1$.</p>

<p>In order for me to price discriminate my profit function needs to comply with some incentive compatibility constraints (I found 6 in total):</p>

<p>$$v_3(q_3)-P_3≥v_3(q_2)-P_2$$</p>

<p>$$v_3(q_3)-P_3≥v_3(q_1)-P_1$$</p>

<p>$$v_2(q_2)-P_2≥v_2(q_3)-P_3$$</p>

<p>$$v_2(q_2)-P_2≥v_2(q_1)-P_1$$</p>

<p>$$v_1(q_1)-P_1≥v_1(q_3)-P_3$$</p>

<p>$$v_1(q_1)-P_1 \geq v_1(q_2)-P_2$$</p>

<p>Where I suppose my profit function is:</p>

<p>$$\pi=P_3+P_2+P_1-c(q_3+q_2+q_1)$$</p>

<p>where c is the constant marginal cost. </p>

<p>How do I deduce the optimal prices?</p>

<p>The problem is I've only seen problems with two types of consumers where the optimal prices were:</p>

<p>$$P_1=v_1(q_1)$$</p>

<p>$$P_3=v_1(q_1)+v_3(q_3)-v_3(q_1)$$</p>

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

CC BY-SA 3.0
2017-04-06T19:40:38.230Z
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
CC BY-SA 3.0
2017-04-07T06:29:36.737Z
Don't you have individual rationality constraints as well? I.e., for instance $P_1 \leq v(q_1)$?
Oliv
CC BY-SA 3.0
2017-04-07T06:37:25.857Z
Yes, but I guess these are more or less implied by the IC constraints. Maybe just except v(q1)-P1>0
user12783
CC BY-SA 3.0
2017-04-07T06:56:49.503Z
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
<microeconomics><industrial-organisation><price-discrimination>
Price discrimination
user12783
# Price discrimination

(Asked by: username_0 on 2017-04-06)

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.

**Tags:** microeconomics, industrial-organisation, price-discrimination

> **Comments:**
> **username_0**: 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.  
> **username_1**: Don't you have individual rationality constraints as well? I.e., for instance $P_1 \leq v(q_1)$?  
> **username_0**: Yes, but I guess these are more or less implied by the IC constraints. Maybe just except v(q1)-P1>0  
> **username_1**: 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?  

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## 0 answers

