# Menu-pricing with three consumer groups

I want to analyze the following setting:

An entrepreneur (with monopoly power) sells a product in two periods. In period 1 there are two consumer groups (denoted by 1 and 2) and in period 2 there is only one consumer group (denoted by 3). He has to set the three prices $p_1$, $p_2$ and $p_3$ in order to maximize his total profit (period 1 and 2), taking into account each consumer's participation and incentive compatibility constraints.

The different consumer groups are separated by their taste parameter, $θ$. A consumer in period 2 derives the following utility: $U_3 = θ_3 - p_3$, where $p$ is the price. The consumers in period 1 receive additional benefits by consuming in period 1. A consumer from group 1 and group 2 get respectively: $U_1 = θ_1(1+δ) - p_1$ and $U_2 = θ_2(1+δβ) - p_2$, where $δ>0$ and $0<β<1$.

I assume the following ranking $θ_1 > θ_2 > θ_3$. Assuming a unit mass of consumers, the consumer groups can be divided like this: $θ ∈ [0,θ_2]$ (period 2 consumers), $θ ∈ [θ_2,θ_1]$ and $[θ_1,1]$ (period 1 consumers). I also assume a unit demand.

I write the consumer's participation (PC) and incentive compatibility constraints (IC) for each consumer group:

Consumer group 3:

$U_3 \geq 0$ (PC3)

$U_3 \geq U_2$ (IC3A)

$U_3 \geq U_1$ (IC3B)

Consumer group 2:

$U_2 \geq 0$ (PC2)

$U_2 \geq U_3$ (IC2A)

$U_2 \geq U_1$ (IC2B)

Consumer group 1:

$U_1 \geq 0$ (PC1)

$U_1 \geq U_3$ (IC1A)

$U_1 \geq U_1$ (IC1B)

The entrepreneur's maximization problem is:

$π = p_1q_1 + p_2q_2 + p_3q_3$

, where $q$ is the output or the number of consumers in each consumer group.

Finding the equilibrium prices that induce self-selection, I solve for the subgame-perfect Nash equilibrium.

In order to solve the game using backwards induction, I have to find the "right" PC's and IC's.

When there are only two consumers groups at least one PC and one IC have to hold. Usually, it is participation of the low type (in this case $θ_3$) and self-selection of the high type (in this case $θ_1$).

If I assume all IC's and PC's bind in equilibrium ($=$ instead of $\geq$), then I get $θ_1 = θ_2$. I then tried to separate the consumer groups according to their type like this:

$θ_1(1+δ) - p_1 = θ_1(1+δβ) - p_2$ (here I want to separate the two consumer groups in period 1)

$θ_2(1+δβ) - p_2 = θ_2 - p_3$ (here I want to separate the low-valuation consumer in period 1 from the consumer in period 2)

This approach didn't induce self-selection between the two consumer groups in period 1. More specifically, this approach is wrong, I believe because the results are $θ_1 = 1/2$ and $θ_2 = \frac{1+2δ}{1+4δβ}$, which violates the condition from earlier: $θ_1 > θ_2 > θ_3$.

Any suggestion on which PC's and IC's should hold in equilibrium in order for self-selection to appear is much appreciated.

• Anyone that has a suggestion? :) Jul 6 '16 at 12:00