Price Discrimination and Revealed Preference

Christopher Makler

Stanford University Department of Economics

 

Econ 51: Lecture 18

Both of these are models of asymmetric information: an uninformed party is trying to extract behavior or information from an informed party.

Last time:
The Principal-Agent Model

Today:
Price Discrimination

  • Buyers differ as to their valuation of quality: some value it a lot, others not so much
  • A seller cannot observe how much the buyer values quality
  • Offers a menu of options: a "budget" product at a low price point, and a "premium" product at a higher price point
  • Goal: have buyers sort themselves
  • A "principal" wants an "agent" to do something for them
  • The principal cannot observe how much effort the agent puts forward, but can observe if the agent is successful
  • Offers a wage contract with two values: one if the agent fails, the other if they succeed
  • Goal: encourage effort

Price Discrimination

  • Neoclassical model: perfect competition, single price, price-taking
  • Real world: firms with market power engage in lots of interesting kinds of pricing strategies
    • Transportation: single tickets vs. monthly passes
    • Cell phone plans: pre-paid vs. unlimited
    • Airline tickets
    • College tuition
  • Asymmetric information problem: the firm doesn't know how much its customers value its product. How can it design different options that encourage customers to self-select based on their preferences?

Different Options for Different Customers

  • The firm is going to have different "offerings" aimed at different customers.
  • One possibility: bundles of quantities
  • Another possibilities: quality choice

Quantity Options

Charge and pay as you go

$1 per point

Rides are 5-8 points each

$109.95 + tax

Unlimited rides through 2023

No blackout dates

Quality Options

Only too often does the sight of third-class passengers travelling in open or poorly sprung carriages,
and always badly seated, raise an outcry against the barbarity of the railway companies.

It wouldn't cost much, people say, to put down a few yards of leather and a few pounds of horsehair, and it is worse than avarice not to do so...

It is not because of the few thousand francs which would have to be spent to put a roof over the third-class carriages or to upholster the third class seats that some company or other has open carriages with wooden benches; it would be a small sacrifice for popularity.

What the company is trying to do is to prevent the passengers who can pay the second-class fare from traveling third class; it hits the poor, not because it wants to hurt them, but to frighten the rich.

- Emile Dupuit, 19th century French railroad engineer

Model Setup

Firm chooses to produce goods with quality \(q\)

Type 1 (low value)

There are two types of consumers, who value quality differently.

Type 2 (high value)

TB_1(q) = 20q - {1 \over 2}q^2
TB_2(q) = 30q - {1 \over 2}q^2

Assume (for now) equal numbers in each group

Assume the firm has no costs; they are just trying to maximize their revenue.

First-Degree Price Discrimination

Type 1 (low value)

Type 2 (high value)

TB_1(q) = 20q - {1 \over 2}q^2
TB_2(q) = 30q - {1 \over 2}q^2

Suppose the firm can observe the type of each customer, and offer them a quality just suited to them — and charge them their total willingness to pay.

p_1(q_1) = TB_1(q_1)
p_2(q_2) = TB_2(q_2)

What qualities will it produce?

What will it charge?

"Budget offering"

"Premium offering"

Second-Degree Price Discrimination

Type 1 (low value)

Type 2 (high value)

TB_1(q) = 20q - {1 \over 2}q^2
TB_2(q) = 30q - {1 \over 2}q^2

Now suppose the firm cannot observe the type of the consumer.

Each consumer will buy the good which gives them the most surplus (benefit minus cost)

p_1(q_1) = TB_1(q_1)
p_2(q_2) = ?

We don't have to worry about the Type-1 consumers buying the premium product

Might the Type-2 consumers want to buy the budget product, though...?

Second-Degree Price Discrimination

Type 1 (low value)

Type 2 (high value)

TB_1(q) = 20q - {1 \over 2}q^2
TB_2(q) = 30q - {1 \over 2}q^2
p_1(q_1) = TB_1(q_1)
TB_2(q_2) - p_2 \ge TB_2(q_1) - p_1

Charge low-value types their maximum willingness to pay:

Constraint for high-value types: prefer to buy \(q_2\) at price \(p_2\) than \(q_1\) at price \(p_1\):

450 - p_2 \ge [30q_1 - {1 \over 2}q_1^2] - [20q_1 - {1 \over 2}q_1^2]
p_1(q_1) = 20q_1 - {1 \over 2}q_1^2
p_2(q_1) = 450 - 10q_1
450 \ge p_2 + 10q_1

Notice: the price you can charge for the premium product depends on how nice the budget product is. The crappier the budget version, the more you can charge for premium...

Second-Degree Price Discrimination

Type 1 (low value)

Type 2 (high value)

p_1(q_1) = 20q_1 - {1 \over 2}q_1^2
p_2(q_1) = 450 - 10q_1

Notice: the price you can charge for the premium product depends on how nice the budget product is. The crappier the budget version, the more you can charge for premium...

Second-Degree Price Discrimination

Type 1 (low value)

Type 2 (high value)

TB_1(q) = 20q - {1 \over 2}q^2
TB_2(q) = 30q - {1 \over 2}q^2
p_1(q_1) = 20q_1 - {1 \over 2}q_1^2
p_2(q_1) = 450 - 10q_1

Expected revenue if equal numbers of each type:

\mathbb{E}[p(q_1)] = {1 \over 2}[20q_1 - {1 \over 2}q_1^2] + {1 \over 2}[450 - 10q_1]

Take the derivative and set equal to zero:

{1 \over 2}[20 - q_1] + {1 \over 2}[- 10] = 0
\Rightarrow q_1^* = 10
q_1^* = 10

Summary

  • In each of the models we saw this week, one of the players designs a choice for the other player
  • Principal-agent: incentivize the other player to behave in a certain  way, even though behavior can't be monitored
  • Price discrimination: incentivize the other player to reveal their preferences by giving them a menu of options
  • How many games in the real world are designed...and by whom...and for what (profit-making) purpose...?

Bonus Content: Revealed Preference

  • Do people actually have "utility functions" that the firms are trying to figure out?
  • Not necessarily: but they may behave as if they do

Paul Samuelson

Utility is taken to be correlative to Desire or Want.

Desires cannot be measured directly, but only indirectly,
by the outward phenomena to which they give rise...

the measure is found in the price which a person is willing to pay for the fulfilment or satisfaction of his desire.

Suppose we don't know anything about someone's "utility function," but we can observe some choices that they make when faced with different choice sets.

Suppose someone chooses some bundle A when bundle B is also an option (\(BL_1\)).

We know that they must therefore at least weakly prefer A to B.

A
B

Now suppose we see another time when A is not an option, but B and C are (\(BL_2\)); and they choose B.

We now know that they must prefer A to C, and everything in that new budget set.

BL_1
BL_2
C

We don't have to take seriously the notion that people “have utility functions," just that their behavior can be modeled as if they do.

Econ 51 | 18 | Price Discrimination and Revealed Preference

By Chris Makler

Econ 51 | 18 | Price Discrimination and Revealed Preference

Designing mechanisms to reveal preferences

  • 279