asymmetric information econ 370: microeconomic theory summer 2004 – rice university stanley...

Asymmetric Information

ECON 370: Microeconomic Theory

Summer 2004 – Rice University

Stanley Gilbert

Econ 370 - Asymmetric Information 2

Asymmetric Information

• Up to now, we have assumed – Everyone is fully informed

– Or equally uninformed

• In many cases one party has economically relevant information that another party does not have

• We term this “Asymmetric Information”

• It can produce economic inefficiency


Example

• Suppose there are two types of umbrellas– Good and Bad

• They cannot be distinguished until after they have been used– Bad umbrellas disintegrate after a little use

• Umbrellas are valued as follows

Good

Bad

Value toConsumer

Cost toProduce

$11.50$14

$11$8


Example: Full Information

• Under full information, only good umbrellas would be sold

• Since the cost to make a bad umbrella exceeds the benefit it provides

Good

Bad

Value toConsumer

Cost toProduce

$11.50$14

$11$8


Example: Asymmetric Information

• If ⅔ umbrellas are good– People are willing to pay:

– ⅔(14) + ⅓(8) = $12 per umbrella

– Which is greater than the cost to produce them

Good

Bad

Value toConsumer

Cost toProduce

$11.50$14

$11$8


Example: Asymmetric Information

• But firms make more money selling bad umbrellas

• If all firms are small, they have incentive to switch to making bad umbrellas

• Once ⅔ of firms make bad umbrellas:– People are willing to pay ⅓(14) + ⅔(8) = $10– Which is less than it costs to make any umbrella

Good

Bad

Value toConsumer

Cost toProduce

$11.50$14

$11$8


Example: Observations

• Under full information we have the efficient result– Total Surplus = $14 – 11.50 = $2.50 per umbrella

• Under Asymmetric information the market collapses– Total surplus = 0


Problems

• Adverse Selection– People with a poor hidden characteristic…

– take advantage of other’s ignorance

– Example: Sick people buying life insurance

• Moral Hazard– People who can take hidden actions…

– take advantage of other’s ignorance

– Example: Making poor umbrellas

– Example: Employees shirking


Adverse Selection Example

• A company offers Health insurance

• Each illness costs $100,000

• Two types of people:– “Healthy”

– “Sick”

– Insurance company cannot distinguish them

• Types of people differ in probability of getting sick and in willingness to pay for insurance


Adverse Selection

• Actuarially Fair insurance charges rates exactly equal to the cost to insure

• A Pooling Equilibrium is one in which everyone is charged the same rates, regardless of type

Type% of

PopulationRisk of Illness

Willingness to Pay

Healthy 90% 1/1000 $200

Sick 10% 1/100 $1,500

Cost to Insure

$100

$1,000


Pooling Equilibrium

• If the insurance company pools everybody, it would charge:– 0.9 × 100 + 0.1 × 1000 = $190

• Which everyone is willing to pay

• So a Pooling Equilibrium exists

Type% of


Willingness to Pay

Healthy 90% 1/1000 $200

Sick 10% 1/100 $1,500

Cost to Insure

$100

$1,000


Example 2

• If the insurance company pools everybody, it would charge: 0.8 × 100 + 0.2 × 1000 = $280

• Which only the sick are willing to pay

• So there is NO Pooling Equilibrium exists

• The insurance company must charge $1,000

Type% of


Willingness to Pay

Healthy 80% 1/1000 $200

Sick 20% 1/100 $1,500

Cost to Insure

$100

$1,000


Observations

• Since the company cannot tell “sick” people from “healthy” people

• It can only charge a single average rate

• Although it would happily insure everyone at fair rates– And people would willing pay those rates

• It cannot because it cannot tell people apart

• Therefore a majority are uninsured


Responses

• Some market players lose as a result of asymmetric information

• So they have developed strategies to (partially) overcome the problem

• Two main strategies– Signalling

– Screening


Screening

• Screening:– is an action taken by the ignorant party

– to determine types of people

• In general,– It is a cost imposed on the “low-value” party

– That the “high-value” parties are unwilling to endure


Screening Example

• Average cost to insure everybody: – 0.5 × 100 + 0.5 × 200 = $150

• Which only the sick are willing to pay

• Since there is no pooling equilibrium, – the insurance company must charge at least $200

Type% of


Willingness to Pay

Cost to Insure

Cost of Physical

Healthy 50% 1/1000 $140 $100 $40

Sick 50% 1/500 $250 $200 $150


Screening Policies

• Suppose the insurance company offers two policies– One for $240 with no restrictions

– One for $100 but you must pass a physical to get it

• Anyone can “pass” the physical– But “sick” people have to bribe the doctor to do it

Type% of


Willingness to Pay

Cost to Insure

Cost of Physical

Healthy 50% 1/1000 $140 $100 $40

Sick 50% 1/500 $250 $200 $150


Equilibrium

• Healthy people– Are unwilling to buy the $240 policy

– But will pay the $100 + $40 to get the other policy

• Sick– Are willing to buy the $240 policy

– Would pay the $100 + $150 for the other policy,

– But, it is more expensive than the original policy

Type% of


Willingness to Pay

Cost to Insure

Cost of Physical

Healthy 50% 1/1000 $140 $100 $40

Sick 50% 1/500 $250 $200 $150


Screening Observations

• The insurance company imposes a requirement– That is more costly for “sick” people to meet

• And so is able to separate out “healthy” from “sick” people– And insure everyone

• Since no one has an incentive to change– This qualifies as a separating equilibrium

• Notice that – Compared to the full-information situation

– This is inefficient, due to the cost of the physical


Signaling

• In several of the example above,– The “low-cost” people stood to gain by being

identifiable

• While Screening is a cost imposed by the ignorant party to identify types

• Signaling is a cost voluntarily adopted by knowledgeable parties to signal their types

• Example: Lemon Model


Lemon Model

• On the used car market

• Two types of cars– Good Cars

– Lemons

• The types are indistinguishable to the buyers

• The market has the following characteristics

Type% of

PopulationValue to Buyer

Value to Seller

Good Cars 50% $2,000 $1,500

Lemons 50% $1,000 $500


Pooling

• Since buyers can’t distinguish the cars in advance– They are willing to pay only

– 0.5 × $2000 + 0.5 × $1000 = $1,500

• All sellers are willing to participate at that price

• So this is a pooling equilibrium

Type% of


Value to Seller

Good Cars 50% $2,000 $1,500

Lemons 50% $1,000 $500


Signaling

• Sellers of good cars would like to signal the quality of their cars

• Since doing so would enable them to charge $2,000

• But, it has to be in a way that sellers of lemons are unwilling to emulate

Type% of


Value to Seller

Good Cars 50% $2,000 $1,500

Lemons 50% $1,000 $500


Inspecting Lemons

• Sellers can submit their cars for inspection and certification

• Profits for owners of good cars with inspection:– 2000 – 200 – 1500 > 1500 – 1500

• So profits from deviating exceed pooling profits

• So there is no longer a pooling equilibrium

Type% of


Value to Seller

Cost to pass inspection

Good Cars 50% $2,000 $1,500 $200

Lemons 50% $1,000 $500 $1,100


Separating Lemons

• Evaluate the separating equilibrium

• Obviously, owners of good cars have no incentive to deviate

• Lemon Owners profits from deviating– 2000 – 1100 – 500 < 1000 – 500

• So Lemon owners will not deviate either

Type% of


Value to Seller

Cost to pass inspection

Good Cars 50% $2,000 $1,500 $200

Lemons 50% $1,000 $500 $1,100


Observations on Signaling

• In our example, owners of good cars have an incentive to deviate from the pooling case– So, the pooling case is not stable

– There is no pooling equilibrium when the inspection regime is available

• On the other hand, no one has an incentive to deviate from the separating case

• The only stable equilibrium here is the separating equilibrium


General Comments

• Different models of this sort may have different outcomes

• All the following are possible– Pooling equilibrium but no separating equilibrium

– Separating equilibrium but no pooling equilibrium

– Both pooling and separating equilibria

– Neither pooling nor separating equilibria


Moral Hazard

• Moral Hazard – The knowledgeable party acts differently…

– than when everyone possesses full information

• Minimizing Moral hazard requires providing incentives to act efficiently

• Example– If I didn’t insure my car, I would install an alarm

– But since it is insured, I do not

– Insurance company’s solution:

– Provide a discount for installing a car alarm


Example: Hiring a CEO

• Our company, YZA Corporation, is hiring a CEO

• Our objective is to maximize profits

• The CEO’s objective is to maximize utility

• Profits depend on the ‘effort’ the CEO exerts

• Effort is costly to the CEO

• Let profits be: Π(e) – w– Where ‘e’ represents ‘effort’

• The CEO’s utility is: U = w – (e)– And can get work elsewhere with utility uA


CEO Roadmap

• We will evaluate the following cases:

• Full-information equilibrium

• Asymmetric information equilibrium– When the CEO is risk-neutral

– When the CEO is risk-averse

• We seek to identify – The optimal amount of effort the CEO should exert

– And an optimal contract to induce that effort


Full-Information CEO

• The CEO must provide the optimal effort willingly

• Thus we have a Participation Constraint– w* – (e*) ≥ uA

• We have no reason to want to pay more, so set– w* – (e*) = uA

• Profit maximization means:( )[ ] ( ) ( )[ ]euewe A

eeφ−−Π=−Π maxmax

• Which implies that the optimal e* satisfies:

( ) ( )** ee φ′=Π′


Full-Information Observations

• An optimal contract would consist of– w* = uA + (e*) if she works e*

– Zero otherwise

• This is exactly what she would work if she owned the company herself– To see this, write an expression for her utility under

those circumstances

• Effort does not need to be directly observable– Since profit is a function only of effort,

– We can determine how much effort was exerted simply be observing profits


Risk-Neutral CEO

• Here let profits be Π(e, ε) – w– Where ε is a random variable

• Since we cannot observe effort– Pay must take the form: w(Π)

• Our Participation Constraint is– E[w(Π(e*, ε))] – (e*) ≥ uA

• Set E[w(Π(e*, ε))] = uA + (e*)

• Profits become( ) ( )( )[ ]{ } ( )[ ] ( )[ ]eueEeweE A

eeφεεε −−Π=Π−Π ,max,,max


Risk-Neutral Contract Requirements

• An optimal contract would satisfy the Incentive Compatibility Constraint

• That is, the Utility maximizing CEO will exert exactly the Profit maximizing effort

• That is, mathematically:– E[w(Π(e*, ε))] – (e*) ≥ E[w(Π(e, ε))] – (e)


Risk-Neutral Contracts

• As before, optimal effort is exactly what she would work if she owned the company

• One optimal contract (then) is to sell her the company

• Another is to allow her to keep any amount above expected profits

• Both have the effect of placing all risk on her– (Since she is risk-neutral, that doesn’t bother her)

• And ensure she makes the optimal decision


Risk-Averse CEO

• We greatly simplify our model for this case

• Two possible states of the world, ε1, ε2

– The states occur with probability p, 1 – p

• Two possible effort levels, e1, e2

• Profits are Π(e1, ε1) = 1– Otherwise, profits are zero

• CEO utility is U = u(w) – (e)– We let (e2) = 0, and (e1) = α, and u(0) = 0– Reservation wage is zero– Wage becomes w1 if profits are 1, w0 otherwise


Risk-Averse CEO AnalysisAssume:

e1 is optimal e2 is optimal

E[u(w(Π(e2, ε)))] – (e2) ≥ uA

or u(w0) – ≥ 0or w0 = 0

Is satisfied by w1 = w0 = 0

Participation Constraint:Participation Constraint:

or

E[u(w(Π(e1, ε)))] – (e1) ≥ uA

pu(w1) + (1 – p)u(w0) – α ≥ 0

Incentive Compatibility:Incentive Compatibility:

E[u(w(Π(e1, ε)))] – (e1) ≥ E[u(w(Π(e2, ε)))] – (e2)

pu(w1) + (1 – p)u(w0) – α ≥ u(w0)

p(u(w1) + u(w0)) ≥ αor


Risk-Averse CEO Solutionif:


w1 = w0 = 0Participation Constraint:pu(w1) + (1 – p)u(w0) = α

Incentive Compatibility:

p[u(w1) – u(w0)] = α

Substituting the latter into the former

u(w0) = 0

So

w0 = 0or

ppuw

αα >⎟⎠

⎞⎜⎝

⎛= −11


Risk-Averse Expected Profits

0 – 0 = 0

w1 = w0 = 0w1 = u-1(α / p)w0 = 0

Expected Profits:Expected Profits:E[Π(e2, ε) – w(Π(e2, ε))]

p(1 – u-1(α / p)) + (1 – p)(0 – 0)

Expected Profits = p(1 – u-1(α / p)) Expected Profits = 0

if:


So, e1 is optimal if 1 ≥ u-1(α / p)


Risk-Averse Observations

• Since w1 > α / p – Profits are reduced from the Risk-Neutral case

– The firm must reimburse the risk-averse CEO for taking on part of the risk

– This amounts to sharing part of the profits with the CEO

– Much like Stock Options

• If u-1(α / p) > 1 > α / p– Then with a risk-Neutral CEO, the optimal amount of effort is e1

– While with a risk-averse CEO, the optimal amount of effort is e2

– In such a case, an inefficient amount of effort is supplied


In General

• Where both principal (the firm) – And agent (the CEO) are risk-neutral

– Then the optimal contract is essentially to sell the firm to the agent

• Where the principal is risk-neutral– And agent is risk-averse

– Then the optimal contract is to pay a portion of profits as an incentive to the agent

– Even still, the result will usually be inefficient

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