eco421: adverse selection

ECO421: Adverse selection

Marcin Pęski

January 21, 2020

Plan

Introduction

Market for lemons

InsuranceFlood insuranceObamacare

Screening with menusMonopolist with price-quality choiceAdverse selectionMenu

Conclusions

Introduction

I Last time, we learned how to model asymmetric information.I Does asymmetric information affect behavior?I One of the simplest cases is adverse selection:

I one agent has a private information,I depending on the information, the agent may choose whether

to trade with another agent,I the self-selection may hurt the other agent.

Introduction

I One of the simplest cases is adverse selection:I two players, Ann and BobI Ann knows the state of the world (Ann’s type),I Bob does not, but his payoffs depend on it.I Bob offers Ann a contract. Ann accepts it or rejects it

depending on her type.I Bob’s payoff from the contract depends on which type of Ann

accept it.Bob needs to take it into account when choosing the contract.

Plan

Introduction

Market for lemons



Conclusions

Market for lemons

ExampleUsed car market:I two types of used cars: good (with probability λ) and bad (i.e.

“lemon”, with prob. 1− λ)I sellers (current owners know the type of the used car)I buyers do not (you can only learn the type after you use it for

some time).

ExampleSimple example:I The value of the car for the buyers is VG = $2000 and

VB = $1000.I The value for the sellers is UG = $1500 and UB = $750.

So, there are gains from trade.

Market for lemonsAll-types trade

I All-type tradeI Suppose that λ = 2

3 . Then, the market can operate at priceP = $1600.

I Indeed, P > UB ,UG so both car owners are happy to trade.I Also, P < λVG + (1− λ)VB = 2

32000 + 131000 = 1666, so the

buyers are happy to trade as well.I Suppose that λ = 1

3 .I Then, P < λVG + (1− λ)VB = $1333 for the buyer to be

happy trading.For instance P = 1300.

I But then, the good car owner does not want to trade!I If the good cars are not on the market, the value of the cars on

the market for the buyer is equal to VB .I But then, the buyers do not want to buy at P > VB .

Market for lemonsAll-types trade

I For what values of λ there is an all-type trade?I For all type trade, there must be a price P such that

P ≥ max (UB ,UG ) = UG

P ≤ λVG + (1− λ)VB ,

which implies that

λVG + (1− λ)VB ≥ max (UB ,UG ) = UG .

I In our example, it is λ ≥ 12 .

Market for lemonsSome (but not all) types trade

I There is always price P such that only bad cars trade on themarket: P ∈ (UB ,VB) .I the buyer knows that the car is bad, so the price must be

smaller than VB ,I the bad seller sells the car at any price above UB .

I There is no price such that only good cars trade.I Why?

Plan

Introduction

Market for lemons



Conclusions

Insurance markets

I Adverse selection is major issue in the insurance markets.I Insurance companies charge fees and pay out compensation in

case of damage.I The fees are small if the overall probability of the damage is

small.I The problem is when only the risky individuals choose to insure

themselves.I Two examples:

I flood insurance,I individual mandate in health insurance markets.

Insurance marketsFlood insurance

ExampleFlood insurance pays out a compensation to homeowners in case offlooding.Most insurance companies do not offer flood insurance. Or theyoffer, but as an add-on, at significantly higher cost. Why?

Insurance marketsFlood insurance

I Suppose that insurance charges fee p and pays outcompensation C in case of damage.

I Homeowner buys insurance if

−p − π (D − C ) + ∆ > −πD,

whereI π is the probability of the damage,I D is the size of the damage,I and ∆ is the risk premium (a value of peace of mind due to

insurance).I Thus, homeowner buys insurance if

p < πC + ∆,

orπ >

1C

(p −∆) .

Insurance marketsFlood insurance: Identical homeowners

I First, suppose that all homeowners have the same probabilityof damage π. Then, they will buy insurance if

p < πC + ∆.

I Insurance companies make profits if

p ≥ πC .

I law of large numbers.

I For the market to function, we need

pboth ∈ (πC , πC + ∆) .

Insurance marketsFlood insurance: Two types of homeowners

I Suppose that there are two types of homeowners πh > πl andsuch that

π = λπh + (1− λ)πl .

I high and low risk types: some homes are simply badly located,or badly graded,

I we assume that the difference is significant:

πh − πl >1λ

∆

C,

which means∆ < λ (πh − πl)C .

I Each homeowner knows its type. λI Insurance companies do not.


I If both types buy insurance, then

p ≥ πC .

I Further

p ≥ πC = (λπh + (1− λ)πl)C

= λπlC + λ (πh − πl)C> λπlC + ∆!

(we are using the fact that differences are significant)I The last inequality means that the price is too high for the

low-risk types.I Market unraveling: Types l won’t buy insurance.I What happens to the price of insurance?


I TBAI If only high risk types buy the insurance, for the insurers to

make profit, we need

ponly h ≥ πhC .I Notice that

ponly h

pboth≥ πhC

πC + ∆

≥ πhC

πC + λ (πh − πl)C

=πh

2λ (πh − πl) + πl

=1

2λ(1− πl

πh

)+ πl

πh

.

I If πlπh

and λ is very small (there are few very high risk types),the price on the “only h” market is much higher than on the“both” market.


I With adverse selection, the low-risk types may stop buyinginsurance.

I That leaves the insurers with only high risk types.I market unraveling.

I Insurers need to jack-up prices.I Result: flood insurance is rare and expensive.

Insurance markets

I How to deal with adverse selection?I One way is to mandate that everybody buys insurance.

I auto insurance (third party liability)

I Universal health coverage in almost all developed countries.I One exception: US.

Insurance markets

I One of the major political debates in US over the last 10 yearsis about the ACA law, more commonly known as Obamacare.

I Prior to Obamacare, about 80% of adults 18-65 had medicalinsurance.I mostly people whose employers provided coverage for them

and their families,I but, large number of employers did not provide insurance,I if you lost your job, and/or got employed by one of the

no-insurance employers, it was difficult to find an affordableinsurance

I it was impossible, if you or your family had a pre-existingcondition.

I Obamacare intended to change it.

Insurance markets

I Three major elements of Obamacare:I insurance companies cannot discriminate with respect to

pre-existing condition,I subsidies for low-income people,I individual mandate: everybody has to buy coverage.

I The last requirement is least popular politically.I But, without it, Obamacare may fall into so-called “death

spiral”.I In Canada, the “individual mandate” is known as the “universal

coverage.” It plays exactly the same role.

Insurance marketsDeath spiral on Obamacare exchanges

I Why the mandate is important?I Without it - market unraveling. Also, known as “death spiral”:

I the healthiest people withdraw from the market →I →prices go up→I →less healthy people withdraw from the market→I →prices go up even higher→I →even less healthy people stay on the market→I →....I →only the most sick people remain on the market.

Insurance marketsDeath spiral on Obamacare exchanges

I The adverse selection story explains the form of the insurancemarket prior to Obamacare.

I The insurers could keep prices lowI either by refusing to insure folks with pre-existing condition,I or by making sure that both healthy and sick people join the

marketI group insurance by employers creates a mini local individual

mandate.

Plan

Introduction

Market for lemons



Conclusions

Screening with menus

I Recall that we consider adverse selection withI two players, Ann and BobI Ann knows the state of the world (Ann’s type),I Bob does not, but his payoffs depend on it.I Bob offers Ann a contract. Ann accepts it or rejects it

depending on her type.I Bob’s payoff from the contract depends on which type of Ann

accept it.I Bob needs to take it into account when choosing the contract.

I Can Bob do anything else?

Screening with menus

I Can Bob do anything else?I Yes.I Bob can try offer a different contract for each of the types.

I menu of contracts,

I When would it work?

Screening with menusMenus of contracts

I Airline tickets:I economy and first class.

I Health insurance:I low fee-high copay vs high fee-low copay,

I Car rentals:I fill the gas and penalty if not vs prepay gas.

I Phone companies.

Price-quality discrimination

ExampleA monopolist sells good at price p and quality q. The cost ofquality is c (q) = 1

2q2.

Two types of consumers: θh (with prob. λ) and θl < θh. Aconsumer has taste for quality θ and buys the good if only if

θq − p ≥ 0.

Price-quality discriminationNo adverse selection

I First, suppose that there is no adverse selection, perhapsbecause the monopolist can distinguish between consumers.

I Then for each type θ, the monopolist chooses (p, q) thatmaximizes profits p − c (q) st.

p ≤ θq.

I monopolist will choose p = θq.I profits

maxqθq − 1

2q2.

I FOC implyq∗ (θ) = θ.

Price-quality discriminationNo adverse selection

I With no adverse selection

p∗ (θ) = θ2,

q∗ (θ) = θ.

I Consumer’s utility

θ∗q (θ)− p∗ (θ) = 0!

I monopolist gets all the consumer’s utility.

Price-quality discrimination

I With adverse selection, there are many possibilities for themonopolist:I design good (p, q) so that both types will buy,I only h types buy,I only l types buy,I nobody buys.I menu: design two goods (pl , ql) and (ph, qh) so that the each

type buys a different good.

Price-quality discriminationBoth types buy

I If both types buy, it must be that

p ≤ θhq, and p ≤ θlq.

I Because the monopolist maximizes profits, it will choosepboth = θlq.

I Profitsp − c (q) = θlq −

12q2.

FOC imply that qboth = θl , and the profits are equal to

θlθl −12

(θl)2 =

12θ2l .

I Consumer l gets 0 utility, but consumer h is better off!

θhqboth − pboth = θhθl − θ2

l = (θh − θl) θl > 0.

Price-quality discriminationOnly type h

I Only type h buys:p ≤ θhq.

I Because the monopolist maximizes profits, it will choosep = θhq.

I Profits

λ (p − c (q)) = λ

(θhq −

12q2).

I notice that the monopolist only serves fraction λ of the market.

I Similar calculations imply that the optimal profits are equal to

λ12θ2h.

Price-quality discriminationOnly type l

I If type l buys, thenp ≤ θlq.

I But then, alsop ≤ θlq,

so type h buys as well.I We have already considered this case.

Price-quality discriminationMenu: Individual rationality

I Two goods (pl , ql) and (ph, qh).I We must have

ph ≤ θhqh, (IRh)

pl ≤ θlql (IRl)

I these conditions are called individual rationality.

Price-quality discriminationMenu: Incentive compatibility

I But also, it’s better be the case that individuals want to buythe goods designed for them, and not the others.

θhqh − ph ≥ θhql − pl , (ICh)

θlql − pl ≥ θlqh − ph. (ICl)

I it’s called incentive compatibility (IC).I If IC is violated, then

I either we will be in the case where both types buy the samegood,

I or the individuals switch.


I IC conditions:

θhqh − ph ≥ θhql − pl (ICh),

θlql − pl ≥ θlqh − ph (ICl).

I Some algebra shows that

θh (qh − ql) ≥ ph − pl ≥ θl (qh − ql) .

I By eliminating the term in the middle, we get

θh (qh − ql) ≥ θl (qh − ql) , or(θh − θl) (qh − ql) ≥ 0.


I Two inequalities:

θh (qh − ql) ≥ ph − pl ≥ θl (qh − ql) ,

(θh − θl) (qh − ql) ≥ 0.

I Because the θh > θl , the second inequality implies that

qh ≥ ql .

Because θl ≥ 0, the first inequalities imply that

ph ≥ pl .

LemmaIf menu satisfies IC, then types h pay more and get higher qualitygoods.

Price-quality discriminationMenu

I Because the monopolist charges the highest possible price, itmust be that

ph = pl + θh (qh − ql) .

I Because IRl condition must hold, we can take

pl = θlql .

I notice that IRh will always hold asph = θlql + θh (qh − ql) = θhqh − (θh − θl) ql < θhqh!

I Profits:

λ (ph − c (qh)) + (1− λ) (pl − c (ql))

=λ

(θlql + θh (qh − ql)−

12q2h

)+ (1− λ)

(θlql −

12q2l

).

Price-quality discriminationMenu

I Monopolist problem

maxqh,ql

λ

(θlql + θh (qh − ql)−

12q2h

)+ (1− λ)

(θlql −

12q2l

).

I FOC wrt ql :

θl − λθh = (1− λ) ql , or

q∗l =θl − λθh1− λ

.

FOC wrt. qh:

λθh = λqh, orqh = θh.

Price-quality discriminationOptimal menu

I To summarize:I consumer l gets

qmenul = θh, p

menul = pmenu

l + θh (qmenuh − qmenu

l ) ,

qmenul =

θl − λθh1− λ

, pmenul = θlq

menul .

I Compare it to the menu from no adverse selection:

q∗h = θh, p∗h = θhq

∗h

q∗l = θl , p∗l = θlq

∗l .


LemmaIN optimal menu,I consumer l gets 0 utility and a good of worse quality than

under no adverse selection.I consumer h gets >0 utility and the same quality good as under

no adverse selection.

I World


I To see it in another way, notice that the no adverse selectionmenu

q∗h = θh, p∗h = θhq

∗h

q∗l = θl , p∗l = θlq

∗l

is no incentive compatibleI there is no problem with IR.

I To see why, notice that if type h chooses (q∗h, p∗h), he gets 0

utility.I But if type h chooses (q∗l , p

∗l ), he gets positive utility:

θhq∗l − p∗l = θhq

∗l − θlq∗l = (θh − θl) q∗l > 0!

I To make sure that h type does not choose the good for type l ,quality of type l good is reduced.


I Airline will on purpose reduce the legroom to incentivize someof us to buy first-class ticket.

I Apple will on purpose put small HD into the Ipad, so thatthose of us who care strongly enough pay higher price for thelarger drives.

I Etc. etc.

Plan

Introduction

Market for lemons



Conclusions

ConclusionsWhat did we learn - concepts

I Adervse selection in lemons market and insuranceI market unraveling.

I Screening with menus.I individual rationality and incentive compatibilityI optimal menus

ConclusionsWhat did we learn - skills

I Find conditions for market unravelling due to adverse selectionI Check whether a menu is incentive compatible and indivdually

rational.I Find optimal menus.

ConclusionsFurther reading

I Market for lemonsI Akerlof, G. (1970), “The market for “lemons”: Quality

uncertainty and the market mechanism,” Quarterly Journal ofEconomics, 84, 488-500.

I Competitive insurance marketI Rothschild, M. and J. Stiglitz (1976), “Equilibrium in

competitive insurance markets: An essay on the economics ofimperfect information,” Quarterly Journal of Economics, 90,629-649.

eco421: adverse selection

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