6 - Monopoly with Price Discrimination

Question 1

What is arbitrage?

Answer 1

Arbitrage is the situation where people take advantage of different buying and selling prices in different markets, to make small amounts of profit. In some financial markets there is a very small margin between the buying price and the selling price.

Question 2

What is first degree price discrimination (and the underlying assumption)?

Answer 2

Producer captures entire surplus of production (consumer+producer surplus)
-> requires perfect information about consumers preferences

Question 3

What is second degree price discrimination (and the underlying assumption)?

Answer 3

Optimal non-linear pricing with imperfect information about consumers’ preferences
-> self selection, menu choices

Question 4

What is third degree price discrimination (and the underlying assumption)?

Answer 4

Direct signal about consumers’ preferences

-> Different consumer groups

Question 5

Explain a first degree price discrimination scenario with one consumer, then extend to n consumers

Answer 5

Unit demand:

• Single consumer with one unit demand
• Willingness to pay: v
• Perfect price discrimination achieved with p=v

Extends to n consumers:

• Different willingness to pay: vi
• Perfect price discrimination achieved with consumer-specific prices pi=vi

Question 6

How can you analyze first degree price discrimination mathematically? What to maximize?

Answer 6

• Two-part tariff (A,p) suffices: T(x)=A+px
• Overall surplus maximized at price pc=MC(xc)
• Linear price pc=MC(xc) leaves consumer surplus
• Ask an “entrance” price A=Sc(membership fee)
  Alternative tariff:
  (B,x) with x=D(pc) and B=Sc+pcD(pc)
  Sc=Rxc0[P(x)¡pc]dx

Question 7

What are the components of the two-part tariff? (monopoly)

Answer 7

Two-part tariff (Aᵢ,pᶜ):

1. Aᵢ: consumer specific “entrance” fee
2. pᶜ: unit price at competitive level pᶜ=MC(xᶜ)

Question 8

What happens to overall welfare when going from no price discrimination to first degree price discrimination?

Answer 8

• improves allocative efficiency
• Raises overall social welfare (sum of consumer + producer surplus)
• > allows Pareto improvements, but requires lump-sum redistribution (right amount is however informationally intensive)

Question 9

What is the set-up for third degree price discrimination?

Answer 9

• Monopolist receives signals to separate consumers in different groups. (E.g. men/women, students/non-students)
• Monopolist can discriminate between groups, no arbitrage between groups
• Monopolist cannot discriminate within group (linear price within a group)
• There are m groups

Question 10

Why is second degree price discrimination difficult for the monopolist and what are solutions?

Answer 10

The monopolist does not have information on which consumer belongs to which group, but would like to charge different prices for different groups.
-> make bundles, different “menus” for the different groups. each group should prefer the menu specifically designed for them.

Question 11

What are incentive and participation constraints in second degre price discrimination?

Answer 11

incentive constraint: each type should prefer their bundle over the other bundle(s)
participation constraint: each type should prefer participating over not participation (Utility >= 0)

Question 12

What are the two ways to find the social optimal price from demand and production costs?

Answer 12

p* = MC

oder maximize W (=total social welfare)

Question 13

What is the lump-sum transfer F in a monopoly?

Answer 13

It is the difference between demand and profit that ensures that the profit is zero. Can be calculated as difference at certain price. Monopolists will use this to make an entrance fee F/n

Question 14

How can you show whether all consumers would be buyers with an entrance fee A*?

Answer 14

The consumers individual surplus has to be larger than the entrance fee.

Question 15

Which two part tariff would a monopolist choose in first degree price discrimination?

Answer 15

CSᵢ = 0

Question 16

How can you find out which of two customers (or both) a monopolist would sell to if price discrimination is not allowed?

Answer 16

1) check which customer has the higher demand (when p is fixed) or is willing to pay the higher price (when demand is fixed)
2) calculate the profit if selling to both customers at the same price
3) calculate the profit if only selling to the high type customer
4) compare the two profits, which is higher?

Question 17

What is the consumer surplus in a two-part tariff (tⱼ, fⱼ) monopoly situation?

Answer 17

uᵢ = CSᵢ(tⱼ) - fⱼ

-> CSᵢ as a function of the demand curve integral between 0 and the “final demand” as a function of tⱼ

Question 18

What are the steps to solve for the optimal tariffs (transfer t / quantity x, transfer f) second-degree price discrimination?

Answer 18

1) find type
2) write down constraints & profit maximization
3) find consumer utilities and variables to plug in (from binding constraints)
4) maximize firm’s profit & solve for all variables to get menu
5) check non-binding constraints

Question 19

When comparing price-quantity combinations and two-part tariffs, why are the latter suboptimal?

Answer 19

A two-part tariff can always be replicated by price-quantity combination such that Dᵢ(tᵢ) = xᵢ and Tᵢ = fᵢ + tᵢ Dᵢ(tᵢ) and satisfies all four constraints, but not the other way around.

Question 20

Why can’t two-part tariffs be replicated with price-quantity combinations? Which conditions would need to be satisfied (and are thus used to disprove it)?

Answer 20

1) quantity demanded must be the same
2) total payment must be the same
3) all PC and IC must be met -> usually, the IC of the high type is not met

Question 21

When will consumers be buyers in perfect price discrimination? How to prove?

Answer 21

Consumers will buy if their individual CS >= A* (the fee). Calculate the individual CS (1/n CS) minus the individual fee.