Christopher Makler
Stanford University Department of Economics
Econ 51: Lecture 8
Part 1:
The Principal-Agent Model
Part 2:
Price Discrimination
Players
Strategies
Payoffs
Players with Hidden Information
Actions
Outcomes
Given this game,
what outcome do we predict will happen?
Given a desired outcome,
what game can we design to achieve it?
"Reverse Game Theory"
The designer is interested in the outcome
but lacks complete information
Selling train tickets to people with different valuations
Auctioning off a painting to people with different (private) valuations
Splitting rent / choosing rooms in an apartment
Hiring an employee/contractor whose effort you can't observe
Selling a car whose type (quality) isn't observable
Fundamental Questions
If people have hidden information,
(there's an adverse selection problem)
what mechanism can a designer establish
to get them to reveal that information?
If people can take hidden actions,
(there's a moral hazard problem)
what mechanism can a designer establish
to get them to choose the action the designer wants them to take?
Principal: Someone who needs someone else to do something
Agent: The person who needs to do the thing
CEO / sales rep
Professor / student
Landowner / farmer
The principal's payoff depends on the actions of the agent
Can they incentivize the agent to do what they want?
I want to hire you to do a project.
You choose whether to exert effort \((e)\):
I can choose a wage stucture \(w\). Payoffs:
Note: "Nature" chooses whether the agent succeeds or fails; but the action the agent takes affects the probabilities
Suppose I can observe your effort.
If I write a contract to require shirking:
I can write a contract that specifies and effort level and a wage if you exert that effort.
If I write a contract to require effort:
Suppose I can observe your effort.
If I write a contract to require shirking:
I can write a contract that specifies and effort level and a wage if you exert that effort.
If I write a contract to require effort:
NOTE: WE WILL ASSUME THAT IF THE AGENT IS INDIFFERENT, THEY'LL CHOOSE WHAT THE PRINCIPAL WANTS
Expected value of project:
Expected payoff: 200 - 100 = 100
Expected value of project:
Expected payoff: 300 - 121 = 179
Now suppose I cannot observe your effort; I can only observe whether or not you succeed.
I can write a contract that specifies a wage level if you succeed \((w_H)\), and one if you fail \((w_L)\).
If you shirk:
If you exert effort:
If you reject the contract: \(u_A(R) = 10, u_P(R) = 0\)
INCENTIVE COMPATIBILITY CONSTRAINT
Exerting effort
should be better than not exerting effort
PARTICIPATION CONSTRAINT
Accepting the contract (and exerting effort)
should be better than rejecting the contract.
If you shirk:
If you exert effort:
If you reject the contract: \(u_A(R) = 10, u_P(R) = 0\)
INCENTIVE COMPATIBILITY CONSTRAINT
Exerting effort
should be better than not exerting effort
PARTICIPATION CONSTRAINT
Accepting the contract (and exerting effort)
should be better than rejecting the contract.
If you shirk:
If you exert effort:
Potential contract: \(w_L = 81\), \(w_H = 144\)
What's the payoff to the principal?
This is worse than the 179 when they could contract under perfect information;
but better than just accepting shirking and getting a payoff of 100.
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
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
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)
Assume (for now) equal numbers in each group
Assume the firm has no costs; they are just trying to maximize their revenue.
Type 1 (low value)
Type 2 (high value)
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.
What qualities will it produce?
What will it charge?
"Budget offering"
"Premium offering"
Type 1 (low value)
Type 2 (high value)
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)
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...?
Type 1 (low value)
Type 2 (high value)
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 \(p_1\):
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...
Type 1 (low value)
Type 2 (high value)
Expected revenue if equal numbers of each type:
Take the derivative and set equal to zero: