Principal-Agent Model and Price Discrimination
Christopher Makler
Stanford University Department of Economics
Econ 51: Lecture 8
Both of these are models of asymmetric information: an uninformed party is trying to extract behavior or information from an informed party.
Part 1:
The Principal-Agent Model
Part 2:
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
Mechanism Design
Game
Players
Strategies
Payoffs
Mechanism
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?
Principal-Agent Model
I want to hire you to do a project.
- If you succeed, it’s worth \(Q^H = 400\) to me
- If you fail, it’s worth \(Q^L = 100\) to me
You choose whether to exert effort \((e)\):
- If you exert effort \((e = 1)\), you succeed with probability \(p = {2 \over 3}\)
- If you shirk \((e = 0)\), you succeed with probability \(p = {1 \over 3}\)
- Exerting effort costs \(c(e) = e\); that is, exerting effort costs \(c = 1\), while shirking costs \(c = 0\).
I can choose a wage stucture \(w\). Payoffs:
- Me (the principal) is Risk-Neutral: \(u_P(Q,w) = Q - w\)
- You (the agent) are Risk-Averse: \(u_A(w,e) = \sqrt{w} - c(e)\)
- If you reject the deal, you get a payoff of \(\underline u = 10\)
Note: "Nature" chooses whether the agent succeeds or fails; but the action the agent takes affects the probabilities
- Success: worth \(Q^H = 400\) to principal
- Failure: worth \(Q^L = 100\) to principal
- Payoff to principal: \(Q - w\)
- Effort: costs \(c=1\), succeed with probability \(p = {2 \over 3}\)
- Shirk: costs \(c=0\), succeed with probability \(p = {1 \over 3}\)
- Payoff to agent: \(\sqrt{w} - c\); outside option \(\underline u = 10\)
Benchmark Model: Contractible 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:
\sqrt{w} - c(0) \ge \underline u
\sqrt{w} - c(0) \ge \underline u
\sqrt{w} - 0 \ge 10
w \ge 100
\sqrt{w} - 1 \ge 10
w \ge 121
- Success: worth \(Q^H = 400\) to principal
- Failure: worth \(Q^L = 100\) to principal
- Payoff to principal: \(Q - w\)
- Effort: costs \(c=1\), succeed with probability \(p = {2 \over 3}\)
- Shirk: costs \(c=0\), succeed with probability \(p = {1 \over 3}\)
- Payoff to agent: \(\sqrt{w} - c\); outside option \(\underline u = 10\)
Benchmark Model: Contractible 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:
w = 100
w = 121
NOTE: WE WILL ASSUME THAT IF THE AGENT IS INDIFFERENT, THEY'LL CHOOSE WHAT THE PRINCIPAL WANTS
Expected value of project:
{1 \over 3} \times 400 + {2 \over 3} \times 100 = 200
Expected payoff: 200 - 100 = 100
Expected value of project:
{2 \over 3} \times 400 + {1 \over 3} \times 100 = 300
Expected payoff: 300 - 121 = 179
- Success: worth \(Q^H = 400\) to principal
- Failure: worth \(Q^L = 100\) to principal
- Payoff to principal: \(Q - w\)
- Effort: costs \(c=1\), succeed with probability \(p = {2 \over 3}\)
- Shirk: costs \(c=0\), succeed with probability \(p = {1 \over 3}\)
- Payoff to agent: \(\sqrt{w} - c\); outside option \(\underline u = 10\)
Unobservable Effort
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 \((w_L)\).
- Success: worth \(Q^H = 400\) to principal
- Failure: worth \(Q^L = 100\) to principal
- Payoff to principal: \(Q - w\)
- Effort: costs \(c=1\), succeed with probability \(p = {2 \over 3}\)
- Shirk: costs \(c=0\), succeed with probability \(p = {1 \over 3}\)
- Payoff to agent: \(\sqrt{w} - c\); outside option \(\underline u = 10\)
If you shirk:
If you exert effort:
u_A(S) = {1 \over 3}\sqrt{w_H} + {2 \over 3}\sqrt{w_L}
u_P(S) = {1 \over 3}(400 - w_H) + {2 \over 3}(100 - w_L)
u_A(E) = {2 \over 3}\sqrt{w_H} + {1 \over 3}\sqrt{w_L} - 1
u_P(E) = {2 \over 3}(400 - w_H) + {1 \over 3}(100 - w_L)
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.
u_A(E) \ge u_A(S)
u_A(E) \ge u_A(R)
If you shirk:
If you exert effort:
u_A(S) = {1 \over 3}\sqrt{w_H} + {2 \over 3}\sqrt{w_L}
u_A(E) = {2 \over 3}\sqrt{w_H} + {1 \over 3}\sqrt{w_L} - 1
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.
u_A(E) \ge u_A(S)
u_A(E) \ge u_A(R)
{2 \over 3}\sqrt{w_H} + {1 \over 3}\sqrt{w_L} - 1 \ge {1 \over 3}\sqrt{w_H} + {2 \over 3}\sqrt{w_L}
{2 \over 3}\sqrt{w_H} + {1 \over 3}\sqrt{w_L} - 1 \ge 10
{1 \over 3}\sqrt{w_H} - {1 \over 3}\sqrt{w_L} \ge 1
\sqrt{w_H} \ge 3 + \sqrt{w_L}
{2 \over 3}(3 + \sqrt{w_L}) + {1 \over 3}\sqrt{w_L} - 1 \ge 10
\sqrt{w_L} \ge 9
w_L \ge 81
\ge 12
w_H \ge 144
- Success: worth \(Q^H = 400\) to principal
- Failure: worth \(Q^L = 100\) to principal
- Payoff to principal: \(Q - w\)
- Effort: costs \(c=1\), succeed with probability \(p = {2 \over 3}\)
- Shirk: costs \(c=0\), succeed with probability \(p = {1 \over 3}\)
- Payoff to agent: \(\sqrt{w} - c\); outside option \(\underline u = 10\)
If you shirk:
If you exert effort:
u_A(S) = {1 \over 3}\sqrt{w_H} + {2 \over 3}\sqrt{w_L}
u_P(S) = {1 \over 3}(400 - w_H) + {2 \over 3}(100 - w_L)
u_A(E) = {2 \over 3}\sqrt{w_H} + {1 \over 3}\sqrt{w_L} - 1
u_P(E) = {2 \over 3}(400 - w_H) + {1 \over 3}(100 - w_L)
Potential contract: \(w_L = 81\), \(w_H = 144\)
What's the payoff to the principal?
u_P(E) = {2 \over 3}(400 - 144) + {1 \over 3}(100 - 81)
= {2 \over 3} \times 256 + {1 \over 3} \times 19
= 177
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.
Where does risk aversion come in?
- In this example: principal is risk neutral, agent is risk averse
- Offering a bonus encourages effort, but the agent doesn't like it because they don't like risk
- If the agent is extremely risk averse, the only possible contract would be a fixed salary (with no incentive to exert effort)
- If the agent approaches risk neutrality, the bonus can be bigger and bigger. In the extreme, the principal could offer the entire 300 ( = 400 - 100) as a bonus, effectively having the agent bear all the risk and all the reward for success.
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)
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 these models, 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...?
Econ 51 | Spring 23 | 8 | Principal-Agent and Price Discrimination
By Chris Makler
Econ 51 | Spring 23 | 8 | Principal-Agent and Price Discrimination
How to design a mechanism to get someone to behave a certain way, or to reveal their true preferences
