Price Discrimination and Mechanism Design
Econ 400
University of Notre Dame
Econ 400
(ND)
Price Discrimination and Mechanism Design
1 / 40
Price Discrimination
There is a seller who offers two types of goods: A high-quality one with
price th , and a low-quality one with price tℓ (but nothing is stopping th
from equalling tℓ ). For example, a car dealership might have both BMWs
and Camrys (Camries?) on the lot.
Econ 400
(ND)
Price Discrimination and Mechanism Design
2 / 40
Price Discrimination
There is a seller who offers two types of goods: A high-quality one with
price th , and a low-quality one with price tℓ (but nothing is stopping th
from equalling tℓ ). For example, a car dealership might have both BMWs
and Camrys (Camries?) on the lot.
There are two types of buyers in the population: High types, who
occur with probability p, and low types, who occur with probability
1 − p.
Econ 400
(ND)
Price Discrimination and Mechanism Design
2 / 40
Price Discrimination
There is a seller who offers two types of goods: A high-quality one with
price th , and a low-quality one with price tℓ (but nothing is stopping th
from equalling tℓ ). For example, a car dealership might have both BMWs
and Camrys (Camries?) on the lot.
There are two types of buyers in the population: High types, who
occur with probability p, and low types, who occur with probability
1 − p.
The high-type’s valuation of the high-quality good is v + h, and of
the low quality good is v + ℓ, with h > ℓ. The low-type’s valuation of
the high-quality good is v , and of the low quality good is v .
Econ 400
(ND)
Price Discrimination and Mechanism Design
2 / 40
Price Discrimination
There is a seller who offers two types of goods: A high-quality one with
price th , and a low-quality one with price tℓ (but nothing is stopping th
from equalling tℓ ). For example, a car dealership might have both BMWs
and Camrys (Camries?) on the lot.
There are two types of buyers in the population: High types, who
occur with probability p, and low types, who occur with probability
1 − p.
The high-type’s valuation of the high-quality good is v + h, and of
the low quality good is v + ℓ, with h > ℓ. The low-type’s valuation of
the high-quality good is v , and of the low quality good is v .
If an agent buys the good, his payoff is his valuation minus the price,
and if he does not buy the good, he gets a payoff of zero. (For
example, a high-type buyer getting the high quality good gets a
payoff v + h − th )
Econ 400
(ND)
Price Discrimination and Mechanism Design
2 / 40
Price Discrimination
There is a seller who offers two types of goods: A high-quality one with
price th , and a low-quality one with price tℓ (but nothing is stopping th
from equalling tℓ ). For example, a car dealership might have both BMWs
and Camrys (Camries?) on the lot.
There are two types of buyers in the population: High types, who
occur with probability p, and low types, who occur with probability
1 − p.
The high-type’s valuation of the high-quality good is v + h, and of
the low quality good is v + ℓ, with h > ℓ. The low-type’s valuation of
the high-quality good is v , and of the low quality good is v .
If an agent buys the good, his payoff is his valuation minus the price,
and if he does not buy the good, he gets a payoff of zero. (For
example, a high-type buyer getting the high quality good gets a
payoff v + h − th )
How should the seller decide on prices to maximize profits?
Econ 400
(ND)
Price Discrimination and Mechanism Design
2 / 40
Implementation
Our goal is to find prices that implement a particular pattern of behavior
by the buyers. In particular, we might want to
Econ 400
(ND)
Price Discrimination and Mechanism Design
3 / 40
Implementation
Our goal is to find prices that implement a particular pattern of behavior
by the buyers. In particular, we might want to
1
Sell low quality goods to all the types.
Econ 400
(ND)
Price Discrimination and Mechanism Design
3 / 40
Implementation
Our goal is to find prices that implement a particular pattern of behavior
by the buyers. In particular, we might want to
1
Sell low quality goods to all the types.
2
Sell high quality goods to all the types.
Econ 400
(ND)
Price Discrimination and Mechanism Design
3 / 40
Implementation
Our goal is to find prices that implement a particular pattern of behavior
by the buyers. In particular, we might want to
1
Sell low quality goods to all the types.
2
Sell high quality goods to all the types.
3
Sell high quality goods to the high type, and low quality goods to the
low type.
Econ 400
(ND)
Price Discrimination and Mechanism Design
3 / 40
Implementation
Our goal is to find prices that implement a particular pattern of behavior
by the buyers. In particular, we might want to
1
Sell low quality goods to all the types.
2
Sell high quality goods to all the types.
3
Sell high quality goods to the high type, and low quality goods to the
low type.
4
(Why won’t “Sell low quality goods to the high type, and high quality
goods to the low type” work?)
Econ 400
(ND)
Price Discrimination and Mechanism Design
3 / 40
Implementation
Our goal is to find prices that implement a particular pattern of behavior
by the buyers. In particular, we might want to
1
Sell low quality goods to all the types.
2
Sell high quality goods to all the types.
3
Sell high quality goods to the high type, and low quality goods to the
low type.
4
(Why won’t “Sell low quality goods to the high type, and high quality
goods to the low type” work?)
So we say things like, “Prices th = 10 and tℓ = 3 implements the high type
buying the high quality good, and the low type buying the low quality
good.”
Econ 400
(ND)
Price Discrimination and Mechanism Design
3 / 40
Individual Rationality
First issue: Buyers can always get a payoff of zero by withdrawing from
the market. This means that if the high type buyer is supposed to buy the
high quality good,
IRh : v + h − th ≥ 0
Econ 400
(ND)
Price Discrimination and Mechanism Design
4 / 40
Individual Rationality
First issue: Buyers can always get a payoff of zero by withdrawing from
the market. This means that if the high type buyer is supposed to buy the
high quality good,
IRh : v + h − th ≥ 0
and if the high type buyer is supposed to buy the low quality good,
IRh′ : v + ℓ − tℓ ≥ 0
Econ 400
(ND)
Price Discrimination and Mechanism Design
4 / 40
Individual Rationality
First issue: Buyers can always get a payoff of zero by withdrawing from
the market. This means that if the high type buyer is supposed to buy the
high quality good,
IRh : v + h − th ≥ 0
and if the high type buyer is supposed to buy the low quality good,
IRh′ : v + ℓ − tℓ ≥ 0
Likewise if the low type buyer is supposed to buy the high quality good,
IRℓ : v − th ≥ 0
and if the low type buyer is supposed to buy the low quality good,
IRℓ′ : v − tℓ ≥ 0
Econ 400
(ND)
Price Discrimination and Mechanism Design
4 / 40
Individual Rationality
First issue: Buyers can always get a payoff of zero by withdrawing from
the market. This means that if the high type buyer is supposed to buy the
high quality good,
IRh : v + h − th ≥ 0
and if the high type buyer is supposed to buy the low quality good,
IRh′ : v + ℓ − tℓ ≥ 0
Likewise if the low type buyer is supposed to buy the high quality good,
IRℓ : v − th ≥ 0
and if the low type buyer is supposed to buy the low quality good,
IRℓ′ : v − tℓ ≥ 0
So every type will have its own IR constraint, saying that, “The seller can’t
ask for more money from the buyer for the good than what it is worth to
buyer.”
Econ 400
(ND)
Price Discrimination and Mechanism Design
4 / 40
Perfect Information
Suppose buyers’ types are observable by the seller. What prices should be
picked?
The high type is willing to pay v + h for the high quality good, v + ℓ
for the low quality good, and 0 for nothing. Setting th = v + h and
offering the high quality good maximizes the seller’s profits from the
high types, and the high types don’t refuse to participate.
Econ 400
(ND)
Price Discrimination and Mechanism Design
5 / 40
Perfect Information
Suppose buyers’ types are observable by the seller. What prices should be
picked?
The high type is willing to pay v + h for the high quality good, v + ℓ
for the low quality good, and 0 for nothing. Setting th = v + h and
offering the high quality good maximizes the seller’s profits from the
high types, and the high types don’t refuse to participate.
The low type is willing to pay v for the high quality good, v for the
low quality good, and 0 for nothing. Setting th = v and offering the
high quality good maximizes the seller’s profits from the low types,
and the low types don’t refuse to participate.
Econ 400
(ND)
Price Discrimination and Mechanism Design
5 / 40
Perfect Information
Suppose buyers’ types are observable by the seller. What prices should be
picked?
The high type is willing to pay v + h for the high quality good, v + ℓ
for the low quality good, and 0 for nothing. Setting th = v + h and
offering the high quality good maximizes the seller’s profits from the
high types, and the high types don’t refuse to participate.
The low type is willing to pay v for the high quality good, v for the
low quality good, and 0 for nothing. Setting th = v and offering the
high quality good maximizes the seller’s profits from the low types,
and the low types don’t refuse to participate.
So when a rich businessperson in a suit and expensive shoes comes to the
dealership, they are shown BMWs and quoted a high price. When a less
wealthy person comes in, the dealer shows them the BMW as well, but
quotes a lower price. Note that the outcome is efficient (since total surplus
is p(v + h) + (1 − p)(v ), which is greater than any other allocation of
goods), and the seller gets all the gains from trade.
Econ 400
(ND)
Price Discrimination and Mechanism Design
5 / 40
Imperfect Information
But what if the buyers privately know their own types? Does offering the
high quality good to the high types and the low types still work?
Econ 400
(ND)
Price Discrimination and Mechanism Design
6 / 40
Imperfect Information
But what if the buyers privately know their own types? Does offering the
high quality good to the high types and the low types still work? The high
type wants to imitate the low type, and pay v instead of v + h, since
v +h−
v
|{z}
Low type price
Econ 400
(ND)
≥v +h−
(v + h)
| {z }
=0
High type price
Price Discrimination and Mechanism Design
6 / 40
Incentive Compatibility
If types are no longer observable, the seller must offer a single price for the
high quality good, and a single price for the low quality good.
Econ 400
(ND)
Price Discrimination and Mechanism Design
7 / 40
Incentive Compatibility
If types are no longer observable, the seller must offer a single price for the
high quality good, and a single price for the low quality good. If we want
to implement the high type buying the high quality good, we need them to
prefer buyer the high quality good at the high quality price to the low
quality good at the low quality price,
Econ 400
(ND)
Price Discrimination and Mechanism Design
7 / 40
Incentive Compatibility
If types are no longer observable, the seller must offer a single price for the
high quality good, and a single price for the low quality good. If we want
to implement the high type buying the high quality good, we need them to
prefer buyer the high quality good at the high quality price to the low
quality good at the low quality price,
ICh : v + h − th ≥ v + ℓ − tℓ
Econ 400
(ND)
Price Discrimination and Mechanism Design
7 / 40
Incentive Compatibility
If types are no longer observable, the seller must offer a single price for the
high quality good, and a single price for the low quality good. If we want
to implement the high type buying the high quality good, we need them to
prefer buyer the high quality good at the high quality price to the low
quality good at the low quality price,
ICh : v + h − th ≥ v + ℓ − tℓ
and we need the low quality type to prefer the low quality good at the low
quality price to the high quality good at the high quality price,
Econ 400
(ND)
Price Discrimination and Mechanism Design
7 / 40
Incentive Compatibility
If types are no longer observable, the seller must offer a single price for the
high quality good, and a single price for the low quality good. If we want
to implement the high type buying the high quality good, we need them to
prefer buyer the high quality good at the high quality price to the low
quality good at the low quality price,
ICh : v + h − th ≥ v + ℓ − tℓ
and we need the low quality type to prefer the low quality good at the low
quality price to the high quality good at the high quality price,
ICℓ : v − tℓ ≥ v − th
Econ 400
(ND)
Price Discrimination and Mechanism Design
7 / 40
Incentive Compatibility
If types are no longer observable, the seller must offer a single price for the
high quality good, and a single price for the low quality good. If we want
to implement the high type buying the high quality good, we need them to
prefer buyer the high quality good at the high quality price to the low
quality good at the low quality price,
ICh : v + h − th ≥ v + ℓ − tℓ
and we need the low quality type to prefer the low quality good at the low
quality price to the high quality good at the high quality price,
ICℓ : v − tℓ ≥ v − th
These are called incentive compatibility constraints.
Econ 400
(ND)
Price Discrimination and Mechanism Design
7 / 40
IR and IC constraints
The individual rationality (IR) and incentive compatibility (IC) constraints
together are actually an extraordinarily powerful idea. In general,
The IR constraints say, “Every type is better off participating the way
the seller wants than dropping out altogether and taking a payoff of
zero”
Econ 400
(ND)
Price Discrimination and Mechanism Design
8 / 40
IR and IC constraints
The individual rationality (IR) and incentive compatibility (IC) constraints
together are actually an extraordinarily powerful idea. In general,
The IR constraints say, “Every type is better off participating the way
the seller wants than dropping out altogether and taking a payoff of
zero”
The IC constraints say, “Every type is better off participating the way
the seller wants than imitating another type”
Econ 400
(ND)
Price Discrimination and Mechanism Design
8 / 40
IR and IC constraints
The individual rationality (IR) and incentive compatibility (IC) constraints
together are actually an extraordinarily powerful idea. In general,
The IR constraints say, “Every type is better off participating the way
the seller wants than dropping out altogether and taking a payoff of
zero”
The IC constraints say, “Every type is better off participating the way
the seller wants than imitating another type”
If all the IR and IC constraints are satisfied, then the outcome the seller
wants will actually be implemented (right?).
Econ 400
(ND)
Price Discrimination and Mechanism Design
8 / 40
The Seller’s Problem
Suppose we want implement the high types buying the high quality good
at price th and the low types buying the low quality goods at price tℓ in
the most profitable way possible for the seller.
Econ 400
(ND)
Price Discrimination and Mechanism Design
9 / 40
The Seller’s Problem
Suppose we want implement the high types buying the high quality good
at price th and the low types buying the low quality goods at price tℓ in
the most profitable way possible for the seller. Then we need to solve:
max pth + (1 − p)tℓ
th ,tℓ
subject to
IRh : v + h − th ≥ 0
IRℓ : v − tℓ ≥ 0
ICh : v + h − th ≥ v + ℓ − tℓ
ICℓ : v − tℓ ≥ v − th
Econ 400
(ND)
Price Discrimination and Mechanism Design
9 / 40
The Seller’s Problem
Suppose we want implement the high types buying the high quality good
at price th and the low types buying the low quality goods at price tℓ in
the most profitable way possible for the seller. Then we need to solve:
max pth + (1 − p)tℓ
th ,tℓ
subject to
IRh : v + h − th ≥ 0
IRℓ : v − tℓ ≥ 0
ICh : v + h − th ≥ v + ℓ − tℓ
ICℓ : v − tℓ ≥ v − th
But what do we do with all these constraints? There are only two prices
and four constraints. Some of the constraints must be equalities — binding
constraints — and some must be strict inequalities — slack constraints.
Econ 400
(ND)
Price Discrimination and Mechanism Design
9 / 40
ICh + IRℓ → IRh
We are going to show that IRh is slack, or irrelevant, since some of the
other constraints already imply it.
Econ 400
(ND)
Price Discrimination and Mechanism Design
10 / 40
ICh + IRℓ → IRh
We are going to show that IRh is slack, or irrelevant, since some of the
other constraints already imply it.
v + h − th ≥ v + ℓ − tℓ
Econ 400
(ND)
Price Discrimination and Mechanism Design
by ICh
10 / 40
ICh + IRℓ → IRh
We are going to show that IRh is slack, or irrelevant, since some of the
other constraints already imply it.
v + h − th ≥ v + ℓ − tℓ
> v − tℓ
Econ 400
(ND)
Price Discrimination and Mechanism Design
by ICh
since l > 0
10 / 40
ICh + IRℓ → IRh
We are going to show that IRh is slack, or irrelevant, since some of the
other constraints already imply it.
v + h − th ≥ v + ℓ − tℓ
> v − tℓ
≥ 0
Econ 400
(ND)
Price Discrimination and Mechanism Design
by ICh
since l > 0
by IRℓ
10 / 40
ICh + IRℓ → IRh
We are going to show that IRh is slack, or irrelevant, since some of the
other constraints already imply it.
v + h − th ≥ v + ℓ − tℓ
> v − tℓ
≥ 0
by ICh
since l > 0
by IRℓ
This means v + h − th > 0, so IRh is slack, and we can drop it.
Econ 400
(ND)
Price Discrimination and Mechanism Design
10 / 40
The Seller’s Problem (2)
We’re left with:
max pth + (1 − p)tℓ
th ,tℓ
subject to
IRℓ : v − tℓ ≥ 0
ICh : v + h − th ≥ v + ℓ − tℓ
ICℓ : v − tℓ ≥ v − th
Econ 400
(ND)
Price Discrimination and Mechanism Design
11 / 40
ICℓ is slack
Suppose ICℓ and IRℓ are both binding, so that
v − tℓ = 0 → tℓ = v
and
v − th = v − tℓ → th = tl
Econ 400
(ND)
Price Discrimination and Mechanism Design
12 / 40
ICℓ is slack
Suppose ICℓ and IRℓ are both binding, so that
v − tℓ = 0 → tℓ = v
and
v − th = v − tℓ → th = tl
Then ICh is v + h − th = h > 0, meaning that the high type strictly prefers
buying the high quality good to the low quality good. The problem here is,
we can now raise the price on the high types and make a higher profit
without the low types dropping out, so this can’t be profit-maximizing.
Basically, ICℓ is an irrelevant constraint that forces the seller to hold back
from really squeezing money from the high types.
Econ 400
(ND)
Price Discrimination and Mechanism Design
12 / 40
The IC /IR Picture
Econ 400
(ND)
Price Discrimination and Mechanism Design
13 / 40
The Seller’s Problem (3)
We’re left with:
max pth + (1 − p)tℓ
th ,tℓ
subject to
IRℓ : v − tℓ = 0
ICh : v + h − th = v + ℓ − tℓ
Econ 400
(ND)
Price Discrimination and Mechanism Design
14 / 40
The Seller’s Problem (3)
We’re left with:
max pth + (1 − p)tℓ
th ,tℓ
subject to
IRℓ : v − tℓ = 0
ICh : v + h − th = v + ℓ − tℓ
Then we must have tℓ∗ = v and th∗ = v + h − ℓ. This gives the seller profits
of p(v + h − l ) + (1 − p)(v ).
Econ 400
(ND)
Price Discrimination and Mechanism Design
14 / 40
Things to notice
The high type gets a payoff of v + h − th = l , and the low type gets a
payoff of v − tℓ = 0. In general, the higher types get higher payoffs,
since they have the option of imitating the low types.
Econ 400
(ND)
Price Discrimination and Mechanism Design
15 / 40
Things to notice
The high type gets a payoff of v + h − th = l , and the low type gets a
payoff of v − tℓ = 0. In general, the higher types get higher payoffs,
since they have the option of imitating the low types.
The high type gets the efficient, high quality good while the low type
gets the inefficient, low quality good. In general, the higher your type
is, the closer you are to getting the efficient quantity.
Econ 400
(ND)
Price Discrimination and Mechanism Design
15 / 40
Things to notice
The high type gets a payoff of v + h − th = l , and the low type gets a
payoff of v − tℓ = 0. In general, the higher types get higher payoffs,
since they have the option of imitating the low types.
The high type gets the efficient, high quality good while the low type
gets the inefficient, low quality good. In general, the higher your type
is, the closer you are to getting the efficient quantity.
Offering anything to the low type is not always profit maximizing. We
could offer the high quality good at a price th = v + h and get profits
of p(v + h), which might be better than p(v + h − l ) + (1 − p)v .
Econ 400
(ND)
Price Discrimination and Mechanism Design
15 / 40
Things to notice
The high type gets a payoff of v + h − th = l , and the low type gets a
payoff of v − tℓ = 0. In general, the higher types get higher payoffs,
since they have the option of imitating the low types.
The high type gets the efficient, high quality good while the low type
gets the inefficient, low quality good. In general, the higher your type
is, the closer you are to getting the efficient quantity.
Offering anything to the low type is not always profit maximizing. We
could offer the high quality good at a price th = v + h and get profits
of p(v + h), which might be better than p(v + h − l ) + (1 − p)v .
Private information gives the buyers more power to bargain against
the seller, but it leads to more inefficiency as the seller combats this
by reducing the quality of the goods offered to some types.
Econ 400
(ND)
Price Discrimination and Mechanism Design
15 / 40
Quality Distortion
The fact that the high type gets the efficient quality and the low type does
not is very important:
“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...
Econ 400
(ND)
Price Discrimination and Mechanism Design
16 / 40
Quality Distortion
The fact that the high type gets the efficient quality and the low type does
not is very important:
“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... What the company is trying to do is 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....
Econ 400
(ND)
Price Discrimination and Mechanism Design
16 / 40
Quality Distortion
The fact that the high type gets the efficient quality and the low type does
not is very important:
“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... What the company is trying to do is 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.... And it is again for the same reason that the companies, having
proved almost cruel to third-class passengers and mean to second-class
ones, become lavish in dealing with first-class passengers.
Econ 400
(ND)
Price Discrimination and Mechanism Design
16 / 40
Quality Distortion
The fact that the high type gets the efficient quality and the low type does
not is very important:
“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... What the company is trying to do is 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.... And it is again for the same reason that the companies, having
proved almost cruel to third-class passengers and mean to second-class
ones, become lavish in dealing with first-class passengers. Having refused
the poor what is necessary, they give the rich what is superfluous.”
Econ 400
(ND)
Price Discrimination and Mechanism Design
16 / 40
Quality distortion
Quantity discounts: The unit price of the large popcorn at the movie
theater is much smaller than the medium and small sizes. Large jugs
of olive oil are much cheaper per ounce than smaller jugs.
Econ 400
(ND)
Price Discrimination and Mechanism Design
17 / 40
Quality distortion
Quantity discounts: The unit price of the large popcorn at the movie
theater is much smaller than the medium and small sizes. Large jugs
of olive oil are much cheaper per ounce than smaller jugs.
Entry-level products are often deficient: Entry level guitars are almost
unnecessarily poorly made (and often really ugly), just to push people
up to the middle-of-the-range guitars.
Econ 400
(ND)
Price Discrimination and Mechanism Design
17 / 40
Quality distortion
Quantity discounts: The unit price of the large popcorn at the movie
theater is much smaller than the medium and small sizes. Large jugs
of olive oil are much cheaper per ounce than smaller jugs.
Entry-level products are often deficient: Entry level guitars are almost
unnecessarily poorly made (and often really ugly), just to push people
up to the middle-of-the-range guitars.
Why do top universities/corporations offer so many amenities, while
lower level ones do not?
Econ 400
(ND)
Price Discrimination and Mechanism Design
17 / 40
Quality distortion
Quantity discounts: The unit price of the large popcorn at the movie
theater is much smaller than the medium and small sizes. Large jugs
of olive oil are much cheaper per ounce than smaller jugs.
Entry-level products are often deficient: Entry level guitars are almost
unnecessarily poorly made (and often really ugly), just to push people
up to the middle-of-the-range guitars.
Why do top universities/corporations offer so many amenities, while
lower level ones do not?
What would these results imply about health care and insurance
provision?
Econ 400
(ND)
Price Discrimination and Mechanism Design
17 / 40
A General Approach
Think carefully about what outcome is trying to be achieved: That
requires deciding the action that you want each type to take.
Econ 400
(ND)
Price Discrimination and Mechanism Design
18 / 40
A General Approach
Think carefully about what outcome is trying to be achieved: That
requires deciding the action that you want each type to take.
Write IR constraints that require every type to be happier taking the
action you want to implement, rather than opting out of the game.
Econ 400
(ND)
Price Discrimination and Mechanism Design
18 / 40
A General Approach
Think carefully about what outcome is trying to be achieved: That
requires deciding the action that you want each type to take.
Write IR constraints that require every type to be happier taking the
action you want to implement, rather than opting out of the game.
Write IC constraints that require every type to be happier taking the
action you want to implement, rather than imitating another type.
Econ 400
(ND)
Price Discrimination and Mechanism Design
18 / 40
A General Approach
Think carefully about what outcome is trying to be achieved: That
requires deciding the action that you want each type to take.
Write IR constraints that require every type to be happier taking the
action you want to implement, rather than opting out of the game.
Write IC constraints that require every type to be happier taking the
action you want to implement, rather than imitating another type.
Write out the seller’s maximization problem, subject to the IR and IC
constraints, and solve the system of inequalities for the optimal prices.
Econ 400
(ND)
Price Discrimination and Mechanism Design
18 / 40
Example: Public Goods
Above, we used our powers for evil, maximizing the profits of sellers. Let’s
look at an example where the IC/IR approach leads to something socially
useful.
Econ 400
(ND)
Price Discrimination and Mechanism Design
19 / 40
Example: Public Goods
Above, we used our powers for evil, maximizing the profits of sellers. Let’s
look at an example where the IC/IR approach leads to something socially
useful.
There are two agents who live along a dirt road outside town. Each
value paving the road at vh with probability p, or vℓ with probability
1 − p. This is privately known to each agent.
Econ 400
(ND)
Price Discrimination and Mechanism Design
19 / 40
Example: Public Goods
Above, we used our powers for evil, maximizing the profits of sellers. Let’s
look at an example where the IC/IR approach leads to something socially
useful.
There are two agents who live along a dirt road outside town. Each
value paving the road at vh with probability p, or vℓ with probability
1 − p. This is privately known to each agent.
The cost of paving the road is c, and the government would like to
pave the road, except that 2vℓ < vh + vℓ < c < 2vh , so that the
paving is only efficient if both agents have high value for it.
Econ 400
(ND)
Price Discrimination and Mechanism Design
19 / 40
Example: Public Goods
Above, we used our powers for evil, maximizing the profits of sellers. Let’s
look at an example where the IC/IR approach leads to something socially
useful.
There are two agents who live along a dirt road outside town. Each
value paving the road at vh with probability p, or vℓ with probability
1 − p. This is privately known to each agent.
The cost of paving the road is c, and the government would like to
pave the road, except that 2vℓ < vh + vℓ < c < 2vh , so that the
paving is only efficient if both agents have high value for it.
The government wishes to determine the agents’ values, and pave
only if it is efficient to do so.
Econ 400
(ND)
Price Discrimination and Mechanism Design
19 / 40
Example: Public Goods
Above, we used our powers for evil, maximizing the profits of sellers. Let’s
look at an example where the IC/IR approach leads to something socially
useful.
There are two agents who live along a dirt road outside town. Each
value paving the road at vh with probability p, or vℓ with probability
1 − p. This is privately known to each agent.
The cost of paving the road is c, and the government would like to
pave the road, except that 2vℓ < vh + vℓ < c < 2vh , so that the
paving is only efficient if both agents have high value for it.
The government wishes to determine the agents’ values, and pave
only if it is efficient to do so.
This is a public goods problem with incomplete information.
Econ 400
(ND)
Price Discrimination and Mechanism Design
19 / 40
Public Goods
Suppose the government sends the agents the following survey: “Tell us a
report of your value, v̂ . If you both say you value the road at vh , the road
will be paved. If either of you has a low value, it won’t be paved.”
Econ 400
(ND)
Price Discrimination and Mechanism Design
20 / 40
Public Goods
Suppose the government sends the agents the following survey: “Tell us a
report of your value, v̂ . If you both say you value the road at vh , the road
will be paved. If either of you has a low value, it won’t be paved.” Is this
incentive compatible? Why or why not?
Econ 400
(ND)
Price Discrimination and Mechanism Design
20 / 40
Public Goods: Individual Rationality
Now suppose the government decides to use taxes. In particular, the
agents face a schedule t(v̂1 , v̂2 ) that assigns a price to each report the
agents might make (not necessarily truthfully: an agent can report
v̂1 6= v1 , or v̂2 6= v2 ).
Econ 400
(ND)
Price Discrimination and Mechanism Design
21 / 40
Public Goods: Individual Rationality
Now suppose the government decides to use taxes. In particular, the
agents face a schedule t(v̂1 , v̂2 ) that assigns a price to each report the
agents might make (not necessarily truthfully: an agent can report
v̂1 6= v1 , or v̂2 6= v2 ). We want to implement the efficient outcome, which
requires implementing truth-telling: Agents report their types honestly,
and the road is paved if both agents report vh , but is not paved otherwise.
Econ 400
(ND)
Price Discrimination and Mechanism Design
21 / 40
Public Goods: Individual Rationality
Now suppose the government decides to use taxes. In particular, the
agents face a schedule t(v̂1 , v̂2 ) that assigns a price to each report the
agents might make (not necessarily truthfully: an agent can report
v̂1 6= v1 , or v̂2 6= v2 ). We want to implement the efficient outcome, which
requires implementing truth-telling: Agents report their types honestly,
and the road is paved if both agents report vh , but is not paved otherwise.
Then our IR constraint for a high type agent is
IRh : p(vh − t(vh , vh )) + (1 − p)(0 − t(vh , vℓ )) ≥ 0
and the IR constraint for the low type agent is
IRℓ : p(0 − t(vℓ , vh ) + (1 − p)(0 − t(vℓ , vℓ )) ≥ 0
Econ 400
(ND)
Price Discrimination and Mechanism Design
21 / 40
Public Goods
Since the payments must be at least zero, the IRℓ constraint implies
IRℓ : − pt(vℓ , vh ) − (1 − p)t(vℓ , vℓ )) ≥ 0
so that t(vℓ , vh ) = t(vℓ , vℓ ) = 0, leaving just the tax if the project is
actually undertaken, t(vh , vh ).
Econ 400
(ND)
Price Discrimination and Mechanism Design
22 / 40
Public Goods: Incentive Compatibility
To implement truth-telling, we need a high type to prefer to report that he
is a high type rather than a low type:
ICh : p(vh − t(vh , vh )) ≥ 0
and a low type to prefer to report he is a low type rather than a high type:
ICℓ : 0 ≥ p(vℓ − t(vh , vh )) + (1 − p)(0)
Econ 400
(ND)
Price Discrimination and Mechanism Design
23 / 40
Public Goods: The Government’s Problem
Then the relevant constraints are
IRh : p(vh − t(vh , vh )) ≥ 0
ICh : p(vh − t(vh , vh )) ≥ 0
ICℓ : 0 ≥ p(vℓ − t(vh , vh )) + (1 − p)(0)
Econ 400
(ND)
Price Discrimination and Mechanism Design
24 / 40
Public Goods: The Government’s Problem
Then the relevant constraints are
IRh : p(vh − t(vh , vh )) ≥ 0
ICh : p(vh − t(vh , vh )) ≥ 0
ICℓ : 0 ≥ p(vℓ − t(vh , vh )) + (1 − p)(0)
Is there a t(vh , vh ) that satisfies these inequalities?
Econ 400
(ND)
Price Discrimination and Mechanism Design
24 / 40
Public Goods: The Government’s Problem
Then the relevant constraints are
IRh : p(vh − t(vh , vh )) ≥ 0
ICh : p(vh − t(vh , vh )) ≥ 0
ICℓ : 0 ≥ p(vℓ − t(vh , vh )) + (1 − p)(0)
Is there a t(vh , vh ) that satisfies these inequalities? Well, any t above vℓ
satisfies the third inequality, and any t less than vh satisfies the first two,
so any vℓ ≤ t ≤ vh will satisfy all three.
Econ 400
(ND)
Price Discrimination and Mechanism Design
24 / 40
Public Goods: The Government’s Problem
Then the relevant constraints are
IRh : p(vh − t(vh , vh )) ≥ 0
ICh : p(vh − t(vh , vh )) ≥ 0
ICℓ : 0 ≥ p(vℓ − t(vh , vh )) + (1 − p)(0)
Is there a t(vh , vh ) that satisfies these inequalities? Well, any t above vℓ
satisfies the third inequality, and any t less than vh satisfies the first two,
so any vℓ ≤ t ≤ vh will satisfy all three. In fact, if we pick t(vh , vh ) = c/2,
the sum of payments exactly covers the cost of the project, so that it is
budget balanced.
Econ 400
(ND)
Price Discrimination and Mechanism Design
24 / 40
Public Goods: Solution
For any vℓ ≤ t ∗ (vh , vh ) ≤ vh , the following game:
If both agents report that their type is vh , the road is paved and they
each pay t(vh , vh ).
If either agent reports that their type is vℓ , the road is not paved that
they pay nothing.
paves the road whenever the project is efficient, implements truth-telling
as a Bayesian Nash equilibrium of the game, and balances the budget if
t(vh , vh ) = c/2.
Econ 400
(ND)
Price Discrimination and Mechanism Design
25 / 40
Public Goods in Strategic Form
Let vrow and vcol be the players’ true private information (either vh or vl ).
We can even write this as a strategic form game:
h
h
L
vrow
c
c
− , vcol −
2
2
0,0
l
0,0
0,0
Note that it is a weakly dominant strategy to be honest.
Econ 400
(ND)
Price Discrimination and Mechanism Design
26 / 40
Public Goods and Mechanism Design
But if
2vℓ < c < vh + vℓ < 2vh
will this still work?
Econ 400
(ND)
Price Discrimination and Mechanism Design
27 / 40
Public Goods and Mechanism Design
But if
2vℓ < c < vh + vℓ < 2vh
will this still work? In general, yes, for markets like this you can always
implement the efficient outcome where honest reporting is a weakly
dominant strategy, but it might not be budget balanced (this is called a
Vickrey-Clarke-Groves mechanism).
Econ 400
(ND)
Price Discrimination and Mechanism Design
27 / 40
Public Goods and Mechanism Design
So, think about what happened in the public goods example:
Econ 400
(ND)
Price Discrimination and Mechanism Design
28 / 40
Public Goods and Mechanism Design
So, think about what happened in the public goods example:
Econ 400
(ND)
Price Discrimination and Mechanism Design
28 / 40
Public Goods and Mechanism Design
So, think about what happened in the public goods example:
We faced a policy-relevant situation where the government was
pursuing an objective that relied on the privately held information of
agents.
Econ 400
(ND)
Price Discrimination and Mechanism Design
28 / 40
Public Goods and Mechanism Design
So, think about what happened in the public goods example:
We faced a policy-relevant situation where the government was
pursuing an objective that relied on the privately held information of
agents.
The efficient outcome was achieved by designing a game where
honestly reporting the privately held information is a Bayesian Nash
Equilibrium.
Econ 400
(ND)
Price Discrimination and Mechanism Design
28 / 40
Public Goods and Mechanism Design
So, think about what happened in the public goods example:
We faced a policy-relevant situation where the government was
pursuing an objective that relied on the privately held information of
agents.
The efficient outcome was achieved by designing a game where
honestly reporting the privately held information is a Bayesian Nash
Equilibrium.
In effect, we are designing markets to achieve outcomes that economic
agents like the government or a seller wishes to implement. This is called
mechanism design, began seriously in the 1980’s, and is one of the most
exciting areas in microeconomics and game theory.
Econ 400
(ND)
Price Discrimination and Mechanism Design
28 / 40
Example: Moral Hazard
The previous examples revolve around private information, but this one
revolves around a hidden action.
Suppose a Board of Directors is trying to incentive a CEO to exert
high effort rather than low effort.
If the CEO exerts low effort, the company makes profits of π with
probability q and profits of 0 with probability 1 − q. If the CEO exerts
high effort, the company makes profits of π with probability p and
profits of 0 with probability 1 − p. Assume p > q, so that effort
makes success more likely.
Exerting effort imposes a private cost on the CEO of e, and exerting
no effort costs nothing. The CEO can always quit and get a payoff of
0 by working for himself.
Econ 400
(ND)
Price Discrimination and Mechanism Design
29 / 40
Example: Moral Hazard
The previous examples revolve around private information, but this one
revolves around a hidden action.
Suppose a Board of Directors is trying to incentive a CEO to exert
high effort rather than low effort.
If the CEO exerts low effort, the company makes profits of π with
probability q and profits of 0 with probability 1 − q. If the CEO exerts
high effort, the company makes profits of π with probability p and
profits of 0 with probability 1 − p. Assume p > q, so that effort
makes success more likely.
Exerting effort imposes a private cost on the CEO of e, and exerting
no effort costs nothing. The CEO can always quit and get a payoff of
0 by working for himself.
If the Board of Directors can only reward the CEO on the basis of the
realized profit (0 or π), not the effort the CEO actually chose, what
compensation schedule should the Board pick?
Econ 400
(ND)
Price Discrimination and Mechanism Design
29 / 40
Moral Hazard: Wages
Then a contract for the CEO is a pair of numbers, w0 and wπ , giving
the CEO’s payoff if the firm’s profits are zero and if the firm’s profits
are π.
Econ 400
(ND)
Price Discrimination and Mechanism Design
30 / 40
Moral Hazard: Wages
Then a contract for the CEO is a pair of numbers, w0 and wπ , giving
the CEO’s payoff if the firm’s profits are zero and if the firm’s profits
are π.
Suppose we wish to implement high effort by the CEO. How do we
achieve this?
Econ 400
(ND)
Price Discrimination and Mechanism Design
30 / 40
Moral Hazard: Individual Rationality
If the CEO can get a payoff of zero by opting out, his Individual
Rationality constraint is
IR :
pw + (1 − p)w0
}
| π {z
−e ≥ 0
CEO’s expected wage, given high effort
Econ 400
(ND)
Price Discrimination and Mechanism Design
31 / 40
Moral Hazard: Incentive Compatibility
If the CEO can always exert low effort rather than high effort and face the
same wage schedule, his Incentive Compatibility constraint is
IC : pwπ + (1 − p)w0 − e ≥ qwπ + (1 − q)w0 − 0
Econ 400
(ND)
Price Discrimination and Mechanism Design
32 / 40
Moral Hazard: The Board’s Problem
Then the Board solves
max p(π − wπ ) + (1 − p)(0 − w0 )
w0 ,wπ
subject to
IR : pwπ + (1 − p)w0 − e ≥ 0
IC : pwπ + (1 − p)w0 − e ≥ qwπ + (1 − q)w0
Econ 400
(ND)
Price Discrimination and Mechanism Design
33 / 40
Moral Hazard: The Board’s Problem
Then the Board solves
max p(π − wπ ) + (1 − p)(0 − w0 )
w0 ,wπ
subject to
IR : pwπ + (1 − p)w0 − e ≥ 0
IC : pwπ + (1 − p)w0 − e ≥ qwπ + (1 − q)w0
Which constraints bind and why?
Econ 400
(ND)
Price Discrimination and Mechanism Design
33 / 40
Moral Hazard: Solution
wπ∗ = e
1−p
p
q
−
1−p 1−q
e
q
1−p
w0∗ = −
q
p
1−q
−
1−p 1−q
Econ 400
(ND)
Price Discrimination and Mechanism Design
34 / 40
Moral Hazard: Solution
wπ∗ = e
1−p
p
q
−
1−p 1−q
e
q
1−p
w0∗ = −
q
p
1−q
−
1−p 1−q
(How can we “fix” the negative w0∗ ?)
Econ 400
(ND)
Price Discrimination and Mechanism Design
34 / 40
Example: Early Adopters
The price of gadgets is usually very high, then drops precipitously as time
goes on. For example, the iPhone was initially $ 500, and is now much
cheaper. People line up to wait to buy Harry Potter books when the
paperback version will be out in a few months at a much lower price. Why
is this?
There are two kinds of buyers: High types with value v + h who occur
with probability p, and low types with value v who occur with
probability 1 − p.
The seller can offer the good at date 0 for a price t0 and date 1 for a
price t1 ( date 0 is the release day).
The buyers and sellers discount payoffs at date 1 by 0 < δ ≤ 1.
Econ 400
(ND)
Price Discrimination and Mechanism Design
35 / 40
Early Adopters: IR/IC Constraints
Suppose we want to implement the following outcome: High types buy the
product on release day at price t0 , and low types buy the product in period
1 at price t1 .
Econ 400
(ND)
Price Discrimination and Mechanism Design
36 / 40
Early Adopters: IR/IC Constraints
Suppose we want to implement the following outcome: High types buy the
product on release day at price t0 , and low types buy the product in period
1 at price t1 . Then the IR constraints are
IRh : v + h − t0 ≥ 0
IRℓ : δ(v − t1 ) ≥ 0
Econ 400
(ND)
Price Discrimination and Mechanism Design
36 / 40
Early Adopters: IR/IC Constraints
Suppose we want to implement the following outcome: High types buy the
product on release day at price t0 , and low types buy the product in period
1 at price t1 . Then the IR constraints are
IRh : v + h − t0 ≥ 0
IRℓ : δ(v − t1 ) ≥ 0
and the IC constraints are
ICh : v + h − t0 ≥ δ(v + h − t1 )
ICℓ : δ(v − t1 ) ≥ v − t0
Econ 400
(ND)
Price Discrimination and Mechanism Design
36 / 40
Early Adopters: The Seller’s Problem
max pt1 + (1 − p)δt2
t0 ,t1
subject to
IRh : v + h − t0 ≥ 0
IRℓ : δ(v − t1 ) ≥ 0
ICh : v + h − t0 ≥ δ(v + h − t1 )
ICℓ : δ(v − t1 ) ≥ v − t0
Econ 400
(ND)
Price Discrimination and Mechanism Design
37 / 40
Early Adopters: The Seller’s Problem
The IRℓ and ICh constraints again bind, giving
max pt0 + (1 − p)δt1
t0 ,t1
subject to
IRℓ : δ(v − t1 ) = 0
ICh : v + h − t0 = δ(v + h − t1 )
Econ 400
(ND)
Price Discrimination and Mechanism Design
38 / 40
Early Adopters: The Seller’s Problem
The IRℓ and ICh constraints again bind, giving
max pt0 + (1 − p)δt1
t0 ,t1
subject to
IRℓ : δ(v − t1 ) = 0
ICh : v + h − t0 = δ(v + h − t1 )
Solving this gives prices of t1∗ = v and t0∗ = v + (1 − δ)h.
Econ 400
(ND)
Price Discrimination and Mechanism Design
38 / 40
Early Adopters: Payoffs
Then the high types get a payoff
v + h − t0∗ = δh
and the low types get a payoff
v − t1∗ = 0
and the firm’s profits are
p(v + (1 − δ)h) + (1 − p)δv
Econ 400
(ND)
Price Discrimination and Mechanism Design
39 / 40
Early Adopters: Price Discrimination and Time
The firm’s profits can also be written
p(v + (1 − δ)h) + (1 − p)δv = v (p + (1 − p)δ) + p(1 − δ)h
Econ 400
(ND)
Price Discrimination and Mechanism Design
40 / 40
Early Adopters: Price Discrimination and Time
The firm’s profits can also be written
p(v + (1 − δ)h) + (1 − p)δv = v (p + (1 − p)δ) + p(1 − δ)h
As δ ↑ 1, this becomes just v : As the buyers become more patient, the
seller’s bargaining power erodes and he sells the goods to everyone at date
zero at a price of v .
Econ 400
(ND)
Price Discrimination and Mechanism Design
40 / 40