Market Design: Adverse Selection - University of Notre...

Market Design: Adverse Selection

1 / 38 Market Design: Adverse Selection

Information

Sellers’ goods have unobservable quality to buyers

Borrowers’ ability to repay is unobservable to banks who lend

Insurees’ health unobservable to insurance companies


Information





Information





The lemons model

There are sellers whose goods are “lemons”, and their quality isq = 0, or “peaches”, and their quality is q = v

The proportion of lemons in the market is p

If a seller keeps their good, they get a payoff sq; if they sell, they getthe price, t

If a buyer makes a purchase, the buyer gets q − t, where q is thegood’s quality and t is the price; if they don’t buy, they get a payoffof zero


The lemons model






The lemons model






The lemons model






The lemons models

We want to know what happens when you include private orincomplete information: how does it impact market efficiency?

The key idea is that being willing to participate in a trade revealsinformation about you


The lemons models

We want to know what happens when you include private orincomplete information: how does it impact market efficiency?

The key idea is that being willing to participate in a trade revealsinformation about you


The lemons model

While the buyer’s payoff is q − t, he doesn’t observe q at the time oftrade: it’s random, like a coin flip with a weighted coin — but heknows that the seller is willing to trade at the prevailing market price.But how should we model the buyer’s payoff?

For bet based on a coin flip, the outcomes are H and T , theprobabilities are p(H) = 1/2 and p(T ) = 1/2, and the utilities to theagent are u(H) and u(T ). The expected utility is

1

2u(H) +

1

2u(T )

For the buyer, the utility of a peach is v − t, the utility of a lemon is0− t, so the expected utility of a purchase is

(1− p)(v − t) + p(0− t) = (1− p)v − t.


The lemons model



1

2u(H) +

1

2u(T )


(1− p)(v − t) + p(0− t) = (1− p)v − t.


The lemons model



1

2u(H) +

1

2u(T )


(1− p)(v − t) + p(0− t) = (1− p)v − t.


(Expected value in general)

To model randomness, we imagine that (1) events occur (lemon orpeach, rain or sunshine, etc), those events have (2) probabilities ofoccuring which (3) determine utilities for the agents

The events are e1, e2, ..., eN , the probability of each event ek is p(ek),and the utility to the agent is u(ek)

Then expected value is given by

E[U] =N∑

k=1

p(ek)u(ek)


(Expected value in general)

Imagine rolling a six-sided die, and getting a dollar for each pip

The events are 1, 2, 3, ..., 6, the probability of each event 1/6, and theutility to the agent is u(1), u(2), ..., u(6)

Then expected value is given by

E[U] =1

6u(1) +

1

6u(2) +

1

6u(3) +

1

6u(4) +

1

6u(5) +

1

6u(6)


The market for lemons

We’ll set the market price equal to the social benefit of the marginaltrade, just like in a price-taking equilibrium (it also sets the payoff ofall buyers to zero)

The price then depends on who participates, which in turn dependson the price itself:

t∗ = E[q|t∗] =

{p0 + (1− p)v , Peaches and lemons traded

0, Only lemons traded


The market for lemons

We’ll set the market price equal to the social benefit of the marginaltrade, just like in a price-taking equilibrium (it also sets the payoff ofall buyers to zero)

The price then depends on who participates, which in turn dependson the price itself:

t∗ = E[q|t∗] =

{p0 + (1− p)v , Peaches and lemons traded

0, Only lemons traded


Trade in the lemons market

There’s two possibilities:

1 The market functions: peaches and lemons trade, t − sv ≥ 0 andt ≥ 0

2 The market “collapses” or “unravels”: only lemons trade, t < sv andt ≥ 0



If the market is functioning, the expected quality of the goods is

p0 + (1− p)v = (1− p)v .

Lemons are obviously willing to trade, but peaches are only willing totrade if (1− p)v ≥ sv , or 1− s ≥ p.

If the market collapses, the expected quality of the goods is 0, so onlylemons trade, and p > 1− s.



If the market is functioning, the expected quality of the goods is

p0 + (1− p)v = (1− p)v .

Lemons are obviously willing to trade, but peaches are only willing totrade if (1− p)v ≥ sv , or 1− s ≥ p.

If the market collapses, the expected quality of the goods is 0, so onlylemons trade, and p > 1− s.



v

p1−s 1

t

t=p0+(1−p)v


Inefficiency

Notice that the outcome is inefficient: since 0 < s < 1, all sellers wishthey could sell

The problem is that making a purchase carries risk, which wouldn’tbe there if sellers could observe quality

The presence of bad goods drives down the market price, destroyingthe gains from trade for peaches

You can think of it as an “informational externality” that justifiesintervention or creates an opportunity for social entrepreneurs


Inefficiency






Inefficiency






Inefficiency






Adverse selection

When the presence of “bad trades” reduces the efficiency of a market, wesay it suffers from adverse selection

The great recession: bad mortgages were mixed with good ones tocreate financial instruments. Once it was learned that the badmortgages were really bad, and risks were not indepedent across themortgages, the market collapsed.

ACA health exchanges: pools of insurees were much less healthy thaninsurers initially believed. After experiencing losses, many insurershave left the exchanges, and many have single insurers left.

Public schools: wealthier families leave public schools forprivate/charter/magnet schools, and the kids remaining have fewerresources, leading more families to leave...


Adverse selection

When the presence of “bad trades” reduces the efficiency of a market, wesay it suffers from adverse selection

The great recession: bad mortgages were mixed with good ones tocreate financial instruments. Once it was learned that the badmortgages were really bad, and risks were not indepedent across themortgages, the market collapsed.

ACA health exchanges: pools of insurees were much less healthy thaninsurers initially believed. After experiencing losses, many insurershave left the exchanges, and many have single insurers left.

Public schools: wealthier families leave public schools forprivate/charter/magnet schools, and the kids remaining have fewerresources, leading more families to leave...


Institutions and Market Design

This leads to two responses by the government, firms, or entrepreneurs:

Signaling: take a costly action to convince others that you are the“good” type (the side with private information acts)

Screening: offer an agent with private information a set of choices toincentivize them into “self-selecting” and revealing their types (theside without private information acts)












Costly Verification

Suppose there are three quality types: low, medium and high.

Low quality goods are worth 0 and happen with probability `, mediumquality goods are worth m and happen with probability pm, and highquality goods happen with probability ph and are worth h, h > m > 0.

If a seller retains his good, he gets a payoff of sq, 0 < s < 1, so thereare gains from trade. If he sells, he gets the price.

If a buyer makes a purchase, she gets the true quality of the goodminus the price.

What are the prices if only low quality goods are sold? Low andmedium? Low and high?

When do only low quality goods trade? Low and medium? All types?

If the high-types pull their goods off the market, can it collapse?


Costly Verification









Costly Verification









Costly Verification









Costly Verification









Costly Verification









Costly Verification

Suppose any seller could pay c to reveal their type and receive thetrue value of their good in the market

When are high types willing to pay for verification? What happens inthe market for low and medium quality goods if high types leave?

If a social entrepreneur invented a technology to reveal type andstarted selling it, what problems might arise?


Costly Verification





Costly Verification





Unintended Consequences and Verification

Statistical discrimination: Ban-the-box legislation

Pre-existing conditions: information destroys opportunities forinsurance

Who monitors the monitors? Moody’s, LIBOR, etc.


Costly Signaling

What if, instead, objective verification wasn’t possible. Can peachesstill separate themselves out?

Suppose there are low-skilled, who occur with probability p, andhigh-skilled workers, who occur with probability 1− p.

College costs c for low-skilled types, but only sc with 0 < s < 1 forhigh-skilled types: they find it easier to get through tough classes

Firms decide what to pay workers based on their degree, not theirtrue type: college graduates get wug and high school grads get whs

High-skilled workers produce profits πH for the firm and low-skilledworkers produce profits πL (if type were observable and labor marketswere perfectly competitive, these would be the wages).


Costly Signaling







Costly Signaling







Costly Signaling







Costly Signaling







Costly Signaling

What must be true for high types to prefer to go to college?

wug − sc ≥ whs

What must be true for low types to prefer to go to high school?

whs ≥ wug − c

Putting these together yields

wug − whs

s≥ c ≥ wug − whs ,

so there is always a pair of wages (wug ,whs) that separate the high-from low-skill workers, even if college does nothing for productivity.This is a “separating equilibrium.”

If the above condition fails because c is too low, it is a “poolingequilibrium” where all types adopt the same behavior.


Costly Signaling


wug − sc ≥ whs


whs ≥ wug − c


wug − whs





Costly Signaling


wug − sc ≥ whs


whs ≥ wug − c


wug − whs





Costly Signaling


wug − sc ≥ whs


whs ≥ wug − c


wug − whs





Costly Signaling

How inefficient is this? The cost c must be weakly greater than thediscounted exected gain in wages from going to college rather thanhigh-school (why has tuition gone up so much over the 20th century?)

What if you are a credit-constrained high-type and can’t afford to payc?

If c is sufficiently high and workers are paid their marginal product,wug = πH and whs = πL. If c is too low (c < wug − whs), however,both types might go to college, and the signal gives no information.Firms then pay expected marginal product,

w̄ = pπH + (1− p)πL

and the value to high types of a college degree falls. Notice thateveryone is still going to college (this is a “rat race”).

What are the incentives for the universities?

Other examples?


Costly Signaling




w̄ = pπH + (1− p)πL



Other examples?


Costly Signaling




w̄ = pπH + (1− p)πL



Other examples?


Costly Signaling




w̄ = pπH + (1− p)πL



Other examples?


Costly Signaling




w̄ = pπH + (1− p)πL



Other examples?


Screening

When the side without private information acts to mitigate theconsequences of adverse selection, it is called screening

We’ll consider the simple case of a monopolist who faces consumerswith different values for a good, and is trying to design the prices fora product line


Price Discrimination

There is a seller who offers two types of goods: A high-quality one withprice th, and a low-quality one with price t` (but nothing is stopping thfrom equalling t`). For example, a car dealership might have both BMWsand Camrys (Camries?) on the lot.

There are two types of buyers in the population: High types, whooccur with probability p, and low types, who occur with probability1− p.

The high-type’s valuation of the high-quality good is v + h, and ofthe low quality good is v + `, with h > `. The low-type’s valuation ofthe 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. (Forexample, a high-type buyer getting the high quality good gets apayoff v + h − th)

How should the seller decide on prices to maximize profits?






























Implementation

Our goal is to find prices that implement a particular pattern of behaviorby 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 thelow type.

4 (Why won’t “Sell low quality goods to the high type, and high qualitygoods to the low type” work?)

So we say things like, “Prices th = 10 and t` = 3 implements the high typebuying the high quality good, and the low type buying the low qualitygood.”


Implementation








Implementation








Implementation








Implementation








Implementation








Individual Rationality

First issue: Buyers can always get a payoff of zero by withdrawing fromthe market. This means that if the high type buyer is supposed to buy thehigh quality good,

IRh : v + h − th ≥ 0

and if the high type buyer is supposed to buy the low quality good,

IR ′h : v + `− t` ≥ 0

Likewise if the low type buyer is supposed to buy the high quality good,

IR` : v − th ≥ 0

and if the low type buyer is supposed to buy the low quality good,

IR ′` : v − t` ≥ 0

So every type will have its own IR constraint, saying that, “The seller can’task for more money from the buyer for the good than what it is worth tobuyer.”24 / 38 Market Design: Adverse Selection



IRh : v + h − th ≥ 0


IR ′h : v + `− t` ≥ 0


IR` : v − th ≥ 0


IR ′` : v − t` ≥ 0




IRh : v + h − th ≥ 0


IR ′h : v + `− t` ≥ 0


IR` : v − th ≥ 0


IR ′` : v − t` ≥ 0




IRh : v + h − th ≥ 0


IR ′h : v + `− t` ≥ 0


IR` : v − th ≥ 0


IR ′` : v − t` ≥ 0


Perfect Information

Suppose buyers’ types are observable by the seller. What prices should bepicked?

The high type is willing to pay v + h for the high quality good, v + `for the low quality good, and 0 for nothing. Setting th = v + h andoffering the high quality good maximizes the seller’s profits from thehigh 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 thelow quality good, and 0 for nothing. Setting th = v and offering thehigh 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 thedealership, they are shown BMWs and quoted a high price. When a lesswealthy person comes in, the dealer shows them the BMW as well, butquotes a lower price. Note that the outcome is efficient (since total surplusis p(v + h) + (1− p)(v), which is greater than any other allocation ofgoods), and the seller gets all the gains from trade (perfect pricediscrimination).25 / 38 Market Design: Adverse Selection

Perfect Information




Perfect Information




Imperfect Information

But what if the buyers privately know their own types? Does offering thehigh quality good to the high types and the low types still work? The hightype wants to imitate the low type, and pay v instead of v + h, since

v + h − v︸︷︷︸Low type price

≥ v + h − (v + h)︸︷︷︸High type price

= 0


Imperfect Information

But what if the buyers privately know their own types? Does offering thehigh quality good to the high types and the low types still work? The hightype wants to imitate the low type, and pay v instead of v + h, since

v + h − v︸︷︷︸Low type price

≥ v + h − (v + h)︸︷︷︸High type price

= 0


Incentive Compatibility

If types are no longer observable, the seller must offer a single price for thehigh quality good, and a single price for the low quality good. If we wantto implement the high type buying the high quality good, we need them toprefer buyer the high quality good at the high quality price to the lowquality good at the low quality price,

ICh : v + h − th ≥ v + `− t`

and we need the low quality type to prefer the low quality good at the lowquality price to the high quality good at the high quality price,

IC` : v − t` ≥ v − th

These are called incentive compatibility constraints.




ICh : v + h − th ≥ v + `− t`


IC` : v − t` ≥ v − th





ICh : v + h − th ≥ v + `− t`


IC` : v − t` ≥ v − th





ICh : v + h − th ≥ v + `− t`


IC` : v − t` ≥ v − th





ICh : v + h − th ≥ v + `− t`


IC` : v − t` ≥ v − th





ICh : v + h − th ≥ v + `− t`


IC` : v − t` ≥ v − th



IR and IC constraints

The individual rationality (IR) and incentive compatibility (IC) constraintstogether are actually an extraordinarily powerful idea. In general,

The IR constraints say, “Every type is better off participating the waythe seller wants than dropping out altogether and taking a payoff ofzero”

The IC constraints say, “Every type is better off participating the waythe seller wants than imitating another type”

If all the IR and IC constraints are satisfied, then the outcome the sellerwants will actually be implemented (right?).














The Seller’s Problem

Suppose we want implement the high types buying the high quality goodat price th and the low types buying the low quality goods at price t` inthe most profitable way possible for the seller. Then we need to solve:

maxth,t`

pth + (1− p)t`

subject toIRh : v + h − th ≥ 0

IR` : v − t` ≥ 0

ICh : v + h − th ≥ v + `− t`

IC` : v − t` ≥ v − th

But what do we do with all these constraints? There are only two pricesand four constraints. Some of the constraints must be equalities — bindingconstraints — and some must be strict inequalities — slack constraints.




maxth,t`

pth + (1− p)t`


IR` : v − t` ≥ 0

ICh : v + h − th ≥ v + `− t`

IC` : v − t` ≥ v − th





maxth,t`

pth + (1− p)t`


IR` : v − t` ≥ 0

ICh : v + h − th ≥ v + `− t`

IC` : v − t` ≥ v − th



ICh + IR` → IRh

We are going to show that IRh is slack, or irrelevant, since some of theother constraints already imply it.

v + h − th ≥ v + `− t`by ICh

> v − t`since l > 0

≥ 0 by IR`

This means v + h − th > 0, so IRh is slack, and we can drop it.


ICh + IR` → IRh

We are going to show that IRh is slack, or irrelevant, since some of theother constraints already imply it.

v + h − th ≥ v + `− t`by ICh

> v − t`since l > 0

≥ 0 by IR`

This means v + h − th > 0, so IRh is slack, and we can drop it.


The Seller’s Problem (2)

We’re left with:maxth,t`

pth + (1− p)t`

subject toIR` : v − t` ≥ 0

ICh : v + h − th ≥ v + `− t`

IC` : v − t` ≥ v − th


IC` is slack

Suppose IC` and IR` are both binding, so that

v − t` = 0→ t` = v

andv − th = v − t` → th = tl

Then ICh is v + h− th = h > 0, meaning that the high type strictly prefersbuying 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 profitwithout the low types dropping out, so this can’t be profit-maximizing.Basically, IC` is an irrelevant constraint that forces the seller to hold backfrom really squeezing money from the high types.


IC` is slack

Suppose IC` and IR` are both binding, so that

v − t` = 0→ t` = v

andv − th = v − t` → th = tl

Then ICh is v + h− th = h > 0, meaning that the high type strictly prefersbuying 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 profitwithout the low types dropping out, so this can’t be profit-maximizing.Basically, IC` is an irrelevant constraint that forces the seller to hold backfrom really squeezing money from the high types.


The IC/IR Picture

��:High type

doesn’t

want to imitate

the low type

��: Low

type doesn’t

drop out

High type

Low typeMedium type doesn’t

want to imitate

the low type

High type

Low type

Low type

doesn’t

drop out

Medium type

High type doesn’t

want to imitate

the medium type

Two Types Three Types




pth + (1− p)t`

subject toIR` : v − t` = 0

ICh : v + h − th = v + `− t`

Then we must have t∗` = v and t∗h = v + h− `. This gives the seller profitsof p(v + h − l) + (1− p)(v).




pth + (1− p)t`

subject toIR` : v − t` = 0

ICh : v + h − th = v + `− t`

Then we must have t∗` = v and t∗h = v + h− `. This gives the seller profitsof p(v + h − l) + (1− p)(v).


Things to notice

The high type gets a payoff of v + h− th = l , and the low type gets apayoff of v − t` = 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 typegets the inefficient, low quality good. In general, the higher your typeis, the closer you are to getting the efficient quantity.

Offering anything to the low type is not always profit maximizing. Wecould offer the high quality good at a price th = v + h and get profitsof p(v + h), which might be better than p(v + h − l) + (1− p)v .

Private information gives the buyers more power to bargain againstthe seller, but it leads to more inefficiency as the seller combats thisby reducing the quality of the goods offered to some types.


Things to notice






Things to notice






Things to notice






Quality Distortion

The fact that the high type gets the efficient quality and the low type doesnot is very important:

“It is not because of the few thousand francs which would have to bespent to put a roof over the third-class carriages or to upholster the

third-class seats that some company or other has open carriages withwooden benches... What the company is trying to do is prevent the

passengers who can pay the second-class fare from traveling third-class; ithits 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, havingproved almost cruel to third-class passengers and mean to second-class

ones, become lavish in dealing with first-class passengers. Having refusedthe poor what is necessary, they give the rich what is superfluous.”


Quality Distortion








Quality Distortion








Quality Distortion








Quality distortion

Quantity discounts: The unit price of the large popcorn at the movietheater is much smaller than the medium and small sizes. Large jugsof olive oil are much cheaper per ounce than smaller jugs.

Entry-level products are often deficient: Entry level guitars are almostunnecessarily poorly made (and often really ugly), just to push peopleup to the middle-of-the-range guitars.

Why do top universities/corporations offer so many amenities, whilelower level ones do not?

What would these results imply about health care and insuranceprovision?


Quality distortion






Quality distortion






Quality distortion






Adverse selection

When agents hold information about themselves that is unobservableto others, it leads to inefficiencies and potential market unraveling

This leads to two potential remedies:

Signaling: the side with private information acts to reveal its type, byverification or costly signalingScreening: the side without private information sets prices or providesincentives for types to reveal their information honestly (e.g., businessversus economy tickets, quantity-discounting, etc.)


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