Customer‘s safety is the future trend of the foodservice and hospitality industry

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Asymmetric Information: Applications

1. Adverse Selection in Insurance Markets

Economics 313

Adverse Selection and Insurance  Last lecture we considered the problem of Moral Hazard

in insurance: people with insurance do not have the correct incentives to undertake the efficient level of care.

 Insurance contracts can also suffer Adverse Selection: “bad” types, or “unsafe” types at least agents facing greater expected costs would happily buy insurance designed for “safe” types, but the reverse is not true.

 Insurers can design so-called “screening” contracts to (partially) overcome this problem.

Adverse Selection and Insurance

 Assume two different risk groups

 Allow insurance companies to offer two different policies: one targeted at the safe types, the other targeted at unsafe types.

 In some circumstances, the choice of contracts will “screen” the types, and help overcome the inefficiencies arising from asymmetric information

Pooling vs. Separating Equilibrium: Refresher

 If we have an equilibrium where all types of buyer choose the same policy, this is known as a pooling equilibrium.  Risk across the two types is pooled.

 If we have an equilibrium where sellers offer two policies, one which is only purchased by safe types and another that is purchased only by unsafe types, this is known as a separating equilibrium.  The policies separate the types.

Pooling vs. Separating Equilibrium: Refresher  We will see that in insurance markets with adverse

selection: 1. There will never exist a pooling equilibrium. 2. There might exist a separating equilibrium.

 Existence depends on number of safe types relative to unsafe types.

 To start thinking about equilibrium in insurance markets with asymmetric info, recall the full info equilibrium.

Notation 𝑈𝑈 = unsafe types 𝑆𝑆 = safe type 𝑊𝑊𝐺𝐺: wealth in good state 𝑊𝑊𝐵𝐵: wealth in bad state 𝐸𝐸Π: Expected Profit of Insurance companies

𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹: (actuarially) Fair Insurance budget line: That is, insurance company charges actuarially fair premium for this type

Assumptions about Wealth  Assume that both types have the same wealth in good

state without insurance, 𝒘𝒘0

 Assume that wealth in the bad state is zero without insurance (you lose everything)

 For some medical conditions, this total loss assumption can be realistic – treatment for some illnesses are extraordinarily expensive, and in the United States medical debt is the cause of many bankruptcies

Assumptions about Insurance Market  Competition between insurance companies drives

expected profits to zero

 If current policies are such that by offering a new policy an insurance company can draw customers away and make expected profit greater than zero, the previous policies cannot be an equilibrium.

Actuarially Fair Premium for Each Type  The fair premium is the expected damages, and must be

higher for unsafe types

 The slope of the budget line in state space for insurance is −(1 − 𝑝𝑝)/𝑝𝑝 where 𝑝𝑝 is the price per dollar of coverage (which must be less than 1) – see Lecture 5.

 This implies types facing a higher premium will have a flatter budget constraint than those with lower premiums. This should make sense to you…. the unsafe types lose more wealth as they move it into the bad state.

Actuarially Fair Premium for Each Type  Assume full information and that safe types are offered

full insurance at premium 𝑤𝑤0 − 𝑋𝑋𝑆𝑆 and unsafe types at 𝑤𝑤0 − 𝑋𝑋𝑈𝑈

wG = wB

FIBLU

FIBLS

XU XS

wB Policy A is the policy offered to safe types under perfect information

Policy B is the policy offered to unsafe types under perfect information

Actuarially Fair Premium for Each Type  Assume full information and that safe types are offered

full insurance at premium 𝑤𝑤0 − 𝑋𝑋𝑆𝑆 and unsafe types at 𝑤𝑤0 − 𝑋𝑋𝑈𝑈

wG = wB

FIBLS

wB Lets assume p is the fair premium for each type.

Remember, under actuarialy fair premiums, risk averse individuals will always purchase full insurance so their utility functions are tangent at the 45 degree line

ICS

XU XS

B FIBLU

ICU

Actuarially Fair Premium for Each Type  Assume full information and that safe types are offered

full insurance at premium 𝑤𝑤0 − 𝑋𝑋𝑆𝑆 and unsafe types at 𝑤𝑤0 − 𝑋𝑋𝑈𝑈

wG = wB

FIBLU

FIBLS ICS

ICU

A BB

wB Remember, under actually fair premiums, risk averse individuals will always purchase full insurance so their utility functions are tangent at the 45 degree line

XU XS w0

Actuarially Fair Premium for Each Type  Assume full information and that safe types are offered

full insurance at premium 𝑤𝑤0 − 𝑋𝑋𝑆𝑆 and unsafe types at 𝑤𝑤0 − 𝑋𝑋𝑈𝑈

wG = wB

FIBLU

FIBLS ICS

wB Question: Could Policies A and B be equilibrium policies with asymmetric information?

A. Yes B. No

XU XS w0

ICU

Actuarially Fair Premium for Each Type  Assume full information and that safe types are offered

full insurance at premium 𝑤𝑤0 − 𝑋𝑋𝑆𝑆 and unsafe types at 𝑤𝑤0 − 𝑋𝑋𝑈𝑈

wG = wB

FIBLU

FIBLS ICS

ICU

wB Question: Could Policies A and B be equilibrium policies with asymmetric information?

A. Yes B. No

XU XS w0

ICU

Actuarially Fair Premium for Each Type  Assume full information and that safe types are offered

full insurance at premium 𝑤𝑤0 − 𝑋𝑋𝑆𝑆 and unsafe types at 𝑤𝑤0 − 𝑋𝑋𝑈𝑈

wG = wB

FIBLU

FIBLS ICS

ICU

wB No: If offered choice of policy A or B, both S and U types would want A (obviously: A offers same coverage, but is cheaper)

XU XS w0

ICU

Actuarially Fair Premium for Each Type  If everyone buys policy A, then E𝜋𝜋< 0 for seller because the

break even on safe types (A is on FIBLS), lose money on unsafe types (A is above FIBLU).

wG = wB

FIBLU

FIBLS ICS

ICU

wB So if both types are going to be

offered the same policy, it must lie between FIBLS and FIBLS in order for E𝜋𝜋 = 0

XU XS w0

ICU

-- end of part 1 --

Asymmetric Information: Applications

2. Pooling Equilibria

Economics 313

Average Risk and Average Fair Premium  How do we calculate exactly where in between the FIBLs?

 It will depend on the number of S types relative to U types.

 Suppose 75% of the population are safe and 25% are unsafe.

 The average level of risk in the population is the probability-weighted average of the different values of 𝜋𝜋𝐵𝐵 (probability of the bad state)

𝜋𝜋𝐵𝐵 𝑃𝑃 =

3 4 𝜋𝜋𝐵𝐵 𝑆𝑆 +

1 4 𝜋𝜋𝐵𝐵 𝑈𝑈

 If we are going to sell one policy to both types, then if this policy has an implicit per dollar premium equal to 𝜋𝜋𝐵𝐵

𝑃𝑃, then 𝐸𝐸𝜋𝜋 = 0

 The pooling policy will lie on a new FIBL, one that corresponds to average risk. That is, FIBLP, which will have slope of −(1 − 𝜋𝜋𝐵𝐵

𝑃𝑃)/𝜋𝜋𝐵𝐵 𝑃𝑃

FIBL For Average Risk  Note that because πB

U > πB P > πB

S , FIBLP lies somewhere between FIBLU and FIBLS.

 The greater the number of safe types relative to unsafe types, the closer FIBLP lies to FIBLS

FIBLU

FIBLS

FIBLP

Why We Can’t Have A Pooling Equilibrium  If there is a pooling equilibrium it must involve both types

buying the same policy and thus will consist of a policy that lies on the 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑝𝑝, suppose this is point A and both types pay “𝑋𝑋𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 for full coverage

FIBLP

wG = wB

𝑋𝑋𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 w0

Why We Can’t Have A Pooling Equilibrium  Note that risk aversion implies strictly convex ICs, but

expected utility theory implies something about the relative slopes of the safe and unsafe ICs

 Assume safe and unsafe types have the same utility function over wealth

FIBLP

wG = wB

𝑋𝑋𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 w0

Why We Can’t Have A Pooling Equilibrium

 MRS for each type = 𝜋𝜋𝐺𝐺𝑀𝑀𝑈𝑈 𝑤𝑤𝐺𝐺 𝜋𝜋𝐵𝐵𝑀𝑀𝑈𝑈 𝑤𝑤𝐵𝐵

 Since 𝜋𝜋𝐵𝐵 𝑈𝑈 > 𝜋𝜋𝐵𝐵

𝑆𝑆 implies that 𝐹𝐹𝐶𝐶𝑈𝑈 through any point is flatter than 𝐹𝐹𝐶𝐶𝑆𝑆 through that same point.

FIBLP

wG = wB

𝑋𝑋𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝

Why We Can’t Have A Pooling Equilibrium  Suppose policy A is offered to both types by firm 1  Can this be an equilibrium? Does anyone have an incentive to

do things differently?  If only policy A is offered both consumers will buy it because it

better than facing risk

FIBLP

wG = wB

𝑋𝑋𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝

On the 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑝𝑝 Firm 1 gets expected profits of zero

But can another firm make another offer and steal some of firm 1’s customers and make a profit?

If so, A can’t be an equilibrium

Why We Can’t Have A Pooling Equilibrium  On the 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑝𝑝 Firm 1 gets expected profits of zero  But can another firm make another offer and steal some of

firm 1’s customers and make a profit?  If so, A can’t be an equilibrium

FIBLP

wG = wB

𝑋𝑋𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝

Why We Can’t Have A Pooling Equilibrium  Suppose firm 2 offers a policy at point B  All safe types prefer B to A so they all switch policies,

leaving firm 1 just selling to unsafe types because unsafe types prefer A to B

FIBLP

wG = wB

𝑋𝑋𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝

B A

Why We Can’t Have A Pooling Equilibrium  A is above the 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑈𝑈 so the expected profit of firm 1 is

now negative and for firm 2, B is below the 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑆𝑆 so expected profit is positive for firm 2

 Since another firm can offer an alternative policy and steal some consumers A can’t be an equilibrium

FIBLP

wG = wB

𝑋𝑋𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝

B AFIBLU

FIBLS

No Pooling Equilibrium Exists  So we know there cannot be a pooling equilibrium at point A.

 In fact, we have actually proven that there cannot be any pooling equilibrium.

 The argument we have made is a very general one, and relies only on the fact that 𝐹𝐹𝐶𝐶𝑈𝑈is flatter than 𝐹𝐹𝐶𝐶𝑆𝑆through any point.

 No matter where we are in this diagram, one policy cannot be sold to both types in equilibrium.

 Any pooled risk policy offered by one firm allows the other firm to offer an alternative policy that steals away just the low risk consumers

-- end of part 2 --

Asymmetric Information: Applications

3. Separating Equilibria

Economics 313

Finding the Separating Equilibrium  So if there is asymmetric information, then we will never have

an equilibrium in which all types are offered the same policy.  That is, there is never a pooling equilibrium.

 What about a separating equilibrium?

 We will be looking at situations where every consumer is offered a choice of two policies, and where all the safe types choose one policy, and all the unsafe types choose the other policy.

 Insurance companies try to design these policies so that each type buys the “right” policy for their type. (Incentive compatibility)

For a Separating Equilibrium  In any separating equilibrium we need the following:

1. Any policy sold to safe types must lie on FIBLS

2. Any policy sold to unsafe types must lie on FIBLU

3. Each consumer must be as happy with the policy they buy as they would be with the alternative policy (or else they would switch policies).

4. There can be no possibility of a competitor insurance company offering an alternative policy that steals some customers and yields E𝜋𝜋 > 0.

 If 1- 4 are satisfied, then the proposed insurance policies are an equilibrium.

 If any statement is not, then the proposed pair cannot be an equilibrium.

Need to have E𝜋𝜋 = 0

Incentive Compatibility  Thus to ensure unsafe types by insurance that is designed

for them, we need to make the policy designed for safe types, less attractive to unsafe types.

 We also need no profitable alternative policies

 This means we need to offer a policy that lies on both FIBLs, but does not induce unsafe types to try and pretend to be safe types

Incentive Compatibility  If there is a separating equilibrium it must be policies

like F and H  Here unsafe types are not going to prefer to be safe

types, but there are no profitable alternative policies

wB FIBLS

Separating Equilibrium  If a separating equilibrium exists, it must be that the

unsafe types are fully insured but the safe types are less than fully insured

 But a separating equilibrium may not exist

 Insurance markets may be inherently unstable

 If a separating equilibrium does exist it is not efficient because the marginal benefit of insurance to the safe types is greater than the marginal cost (slope of IC is greater than slope of budget line)

Summary  In insurance markets with asymmetric information:

 There is never a pooling equilibrium.  There may be a separating equilibrium, as long as the

proportion of safe types is not too high.

 If there is a separating equilibrium, it will be characterized by

 Full insurance for unsafe types.  Less than full insurance for safe types  Inefficiency

Dealing with the Adverse Selection Problem  Policies to correct the market failure?

 Government provision of insurance is one possibility.

 Essentially the government can mandate that it be the only insurance provider, then provide one policy to everybody.

 Example: Basic health care in Canada.  We all receive the same basic level of coverage, and it is (more or

less) impossible to go outside the system for this coverage.  That is, no competitor can steal away just the healthy consumers

from the public system.  It would be profitable to do this, but insurers are (more or less)

prevented from it by law.

What I Expect You to Know  Why a pooling equilibrium does not exist in insurance

markets with asymmetric information with safe and unsafe types

 What must hold for a separating equilibrium to exist

 This is a source of market failure and be able to suggest ways to get around it

-- end of part 3 --

Asymmetric Information: Applications

4. Principal Agent Models

Economics 313

Moral Hazard

 We will now look at an application of moral hazard in labour markets

 Recall that moral hazard is due to asymmetric information after contracts are signed. In this example, the information is about worker effort

 Once a worker is hired, what incentives do they have to maximize their employer’s profit?

 Could be about workers monitoring their employers.

The Principal – Agent Problem

Slide 42

 We will now investigate how the principal – the person who has somebody else work on their behalf – can design a contract that makes the agent – the person who works on behalf of the principal - behave in the way they want to

 From the point of view of the principal, they want to pay the least amount to the agent to work on their behalf and therefore extract the highest possible payoff for themselves.

The Optimal Contract

Slide 43

 We distinguish two cases

1. When effort is observable

2. When effort is unobservable, but correlated with output.

Example

Slide 44

 Assume the agent’s utility is given by 𝑢𝑢 = 𝑤𝑤 − 𝑐𝑐 𝑒𝑒 , where 𝑐𝑐(𝑒𝑒) is the cost associated with effort

 The agent can either exert high effort 𝑒𝑒ℎ or low effort 𝑒𝑒𝑝𝑝

 The agent has an outside option of earning $15,000 without exerting any effort

 The cost of working hard is equal to 𝑐𝑐 𝑒𝑒ℎ = 5000 and the cost of “shirking” is 𝑐𝑐 𝑒𝑒𝐿𝐿 = 1000

Example

 If the agent works hard there is a 70% chance that the agent produces high output ($50,000 in revenues to the principal) and a 30% chance that the agent produces low output ($20,000 in revenues to the principal).

 If the agent shirks, the principal will earn $50,000 with a 50% chance and $20,000 otherwise

 Expected output increases from effort is $6000 vs cost increase to worker of $4000

Example

Slide 46

 The principal wants to maximize profits which is equal to expected revenues minus any payments to the agent

 The principal’s profit is given by 𝑅𝑅 𝑒𝑒 − 𝑤𝑤  R stands for expected revenues of the principal given effort level 𝑒𝑒 of the agent

 w stands for the payment to the agent  This payment may depend on the effort level if it is observable  This payment may depend on the output level if effort is

unobservable

The Case Where Effort is Observable

Slide 47

 First assume that the agent can exert high or low effort and the principal can observed this

 We can first find out what has to be the minimum payment to the agent for each degree of effort to 1. join the firm (the participation constraint) 2. exert a certain effort level (the incentive compatibility

constraint)

 We can then find out whether it is more profitable to induce high effort or low effort from the principal’s perspective

The Minimum Wage for High Effort

Slide 48

 The wage scheme (𝑤𝑤ℎ if effort is high, 𝑤𝑤𝑝𝑝 if effort is low) must satisfy

• the participation constraint (PC) i.e. they will want to join the firm →

𝑤𝑤ℎ − 𝑐𝑐(𝑒𝑒) ≥ 15,000 • the incentive compatibility constraint (ICC) i.e. will want to

exert high effort at the high wage → 𝑤𝑤ℎ − 5,000 ≥ 𝑤𝑤𝑝𝑝 − 1,000

 From PC, the lowest possible wage the principal has to pay the agent to exert high effort is 20,000; to get low effort, must pay at least 16,000

ICC’s for High and Low Effort

Slide 49

 To induce high effort, the contract could say that if the agent does not exert high effort, they get paid less than 16,000, for example 15,000. In the case of high effort, they would be paid $20,000.

Show that in this case it is in the worker’s best interest to exert high effort: From the ICC

𝑤𝑤ℎ − 5,000 ≥ 𝑤𝑤𝑝𝑝 − 1,000

20,000 − 5,000 > 15,000 – 1,000

ICC’s for High and Low Effort

Slide 50

 To induce low effort, the contract could say that if the agent exerts high effort, they get paid less than 20,000, for example 19,000.

Show that in this case it is in the worker’s best interest to exert low effort. From the ICC

𝑤𝑤ℎ − 5,000 ≥ 𝑤𝑤𝑝𝑝 − 1,000

16,000 − 1,000 > 19,000 – 5,000

Which Contract Would Maximize Profits?

Slide 51

 Which contract will the principal offer?

 When exerting high effort (𝑤𝑤ℎ = 20,000, 𝑤𝑤𝑝𝑝 = 15,000) profit is 0.7 ∗ 50,000 + 0.3 × 20,000 − 20,000 = 21,000

 When exerting low effort (𝑤𝑤ℎ = 19,000, 𝑤𝑤𝑝𝑝 = 16,000) profit is 0.5 ∗ 50,000 + 0.5 × 20,000 − 16,000 = 19,000

 Thus the principal maximizes profits if they pays the agent $20,000 conditional on them exerting high effort and $15,000 conditional on him exerting low effort.

-- end of part 4 --

Asymmetric Information: Applications

5. Unobservable Effort

Economics 313

When Actions are Unobservable

Slide 54

 Suppose now that the effort level from the agent in the previous example is not observable. The principal only knows that either revenues were $50,000 or revenues were $20,000. What is the optimal contract for the principal now?

 Suppose the principal pays now a different wage not dependent on the effort level (it is not observable) but dependent on whether the good outcome (high revenues) (𝑤𝑤𝐺𝐺) or the bad outcome (low revenues) (𝑤𝑤𝐵𝐵) occurred.

Iso-profit lines

Slide 55

 For each of the effort levels, we can construct an iso-profit line in wage space. That is, combinations of the wage in the high output state, 𝑤𝑤𝑔𝑔, and wages in low output state, 𝑤𝑤𝐵𝐵 that yield the same expected profit, given the effort choice of the worker.

 When effort is high, the expected revenue is 𝑅𝑅 𝑒𝑒ℎ = 41,000, and the isoprofit line is

𝐸𝐸𝜋𝜋(ℎ) = 41,000 − 0.70𝑤𝑤𝐺𝐺 − 0.30𝑤𝑤𝐵𝐵 Or

𝑤𝑤𝐺𝐺 = 410000

7 − 10𝐸𝐸𝜋𝜋 ℎ

7 −

3 7 𝑤𝑤𝐵𝐵

Similarly, 𝑤𝑤𝐺𝐺 = 70000 − 2𝐸𝐸𝜋𝜋 𝑙𝑙 − 𝑤𝑤𝐵𝐵

PC’s and ICC’s if Action is Hidden

Slide 56

 The participation constraint and the incentive compatibility constraint for the agent if the principal wants to induce high effort are

 PC: 0.7𝑤𝑤𝐺𝐺 + 0.3𝑤𝑤𝐵𝐵 − 5,000 ≥ 15,000

 ICC:

0.7𝑤𝑤𝐺𝐺 + 0.3𝑤𝑤𝐵𝐵 − 5,000 ≥ 0.5𝑤𝑤𝐺𝐺 + 0.5𝑤𝑤𝐵𝐵 − 1,000

PC’s and ICC’s if Action is Hidden

 The participation constraint and the incentive compatibility constraint for the agent if the principal wants to induce low effort are

 PC: 0.5𝑤𝑤𝐺𝐺 + 0.5𝑤𝑤𝐵𝐵 − 1,000 ≥ 15,000

 ICC:

0.5𝑤𝑤𝐺𝐺 + 0.5𝑤𝑤𝐵𝐵 − 1,000 ≥ 0.7𝑤𝑤𝐺𝐺 + 0.3𝑤𝑤𝐵𝐵 − 5,000

PC’s and IC’s if Action is Hidden

Slide 58

 PC for high effort can be rewritten as:

0.7𝑤𝑤𝐺𝐺 + 0.3𝑤𝑤𝐵𝐵 − 5,000 ≥ 15,000

0.7𝑤𝑤𝐺𝐺 ≥ 15,000 + 5,000 − 0.3𝑤𝑤𝐵𝐵

𝑤𝑤𝐺𝐺 ≥ 200,000

7 −

3 7 𝑤𝑤𝐵𝐵

 ICC for high effort can be rewritten as:

0.7𝑤𝑤𝐺𝐺 + 0.3𝑤𝑤𝐵𝐵 − 5,000 ≥ 0.5𝑤𝑤𝐺𝐺 + 0.5𝑤𝑤𝐵𝐵 − 1,000

0.2𝑤𝑤𝐺𝐺 ≥ 0.2𝑤𝑤𝐵𝐵 + 4,000

𝑤𝑤𝐺𝐺 ≥ 𝑤𝑤𝐵𝐵 + 20,000

Diagram: High Effort

Slide 59

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wG = 200,000/7 –(3/7)wB

𝑤𝑤𝐺𝐺

𝑤𝑤𝐵𝐵

wG = wB + 20,000

Diagram: High Effort

Slide 60

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wG ≥ 200,000/7 –(3/7)*wB

PC satisfied

𝑤𝑤𝐺𝐺

Diagram: High Effort

Slide 61

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𝑤𝑤𝐺𝐺

𝑤𝑤𝐵𝐵

wG ≥ wB + 20,000 ICC satisfied

Diagram: High Effort

Slide 62

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ICC & PC satisfied

Isoprofit lines

Profit increases in this direction

𝑤𝑤𝐺𝐺

Diagram: High Effort

Slide 63

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ICC & PC satisfied Thick red line: optimal

contracts; firm maximizes expected profits given ICC and PC.

𝑤𝑤𝐺𝐺

PC’s and IC’s if Action is Hidden

Slide 64

 PC for low effort can be rewritten as: 0.5𝑤𝑤𝐺𝐺 + 0.5𝑤𝑤𝐵𝐵 − 1,000 ≥ 15,000 0.5𝑤𝑤𝐺𝐺 ≥ 15,000 + 1,000 − 0.5𝑤𝑤𝐵𝐵

𝑤𝑤𝐺𝐺 ≥ 160,000/5 − 𝑤𝑤𝐵𝐵

 ICC for low effort can be rewritten as:

0.5𝑤𝑤𝐺𝐺 + 0.5𝑤𝑤𝐵𝐵 − 1,000 ≥ 0.7𝑤𝑤𝐺𝐺 + 0.3𝑤𝑤𝐵𝐵 − 5,000 0.2𝑤𝑤𝐺𝐺 ≤ 0.2𝑤𝑤𝐵𝐵 + 4,000

𝑤𝑤𝐺𝐺 ≤ 𝑤𝑤𝐵𝐵 + 20,000

Diagram: Low Effort

Slide 65

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wage/bad

wage good

wage/bad

wage good

ICC & PC satisfied for inducing low effort

Isoprofit lines

𝑤𝑤𝐺𝐺

35302520151050

wage/bad

wage good

wage/bad

wage good

Diagram: Low Effort

Slide 66

ICC & PC satisfied for inducing low effort

Thick red line: optimal contracts; firm maximizes expected profits given ICC and PC.

𝑤𝑤𝐺𝐺

Optimal Contract

Slide 67

 The minimum wage scheme that guarantees PC means that the expected payment to the agent needs to be 20,000 to induce high effort and 16,000 to induce low effort

 What are expected profits for the principal in each case? Which level of effort does she want to induce?

Induce High Effort

Slide 68

 With high effort:  0.7 × 50,000 + 0.3 × 20,000 − 20,000 = 21,000

 With low effort:  0.5 × 50,000 + 0.5 × 20,000 − 16,000 = 19,000

 Thus the principal is better off inducing high effort.

Examples of Optimal Contracts

Slide 69

 Verify that if the principal pays 27,500 to the agent if the good outcome occurs and 2,500 if the bad outcome occurs, that the agent will participate and exert high effort

 PC: 0.7*27,500 + 0.3*2,500 = 19,250 + 750 ≥ 20,000

 ICC: 0.7*27,500 + 0.3*2,500 – 5,000 ≥ 0.5*27,500 + 0.5*2,500 – 1,000

Other Combinations Work Too

Slide 70

 Note that this is not the only combination that works

 Verify that if the principal pays 26,000 to the agent if the good outcome occurs and 6,000 if the bad outcome occurs, that the agent will participate and exert high effort  In this case the agent is actually indifferent between

exerting high effort and low effort

Non-linear wage contract: an aside

Slide 71

 In the example above, there are only two outcomes and effort shifts the weight in the probability distribution to the higher outcome. Since to encourage effort wages must increase in the good state, in this sense the wage contract is “linear”

 Many other changes to output are consistent with the idea that effort increases output.

 For example, some states could be fully revealing: output levels that only occur under high or low effort.

 This can lead to non-linear wage schedules, where intermediate output levels get higher wages that high output levels.

Non-linear wage contract: an aside

Slide 72

 For example, assume there are three possible outcomes, 20000, 40000, and 50000, where effort changes the probabilities as follows:

 Now any time the principal sees an output of $40000, it must be effort was high. This can lead to wage contracts where the wage is higher for the intermediate output than the high output.

20000 40000 50000

Low effort .5 .5

High effort .2 .1 .7

Conclusion of Principal-Agent Problem

Slide 73

 If workers are paid a lump-sum wage independent of the amount of output, they will not exert effort.

 Paying a lump-sum wage and monitoring is one possibility but monitoring may be costly in itself.

 If effort is unobservable but correlated with output, writing a contract that pays a different wage depending on the observed output level may induce the optimal amount of effort.  Efficient in case of risk neutral principals and agents, but

inefficient in case of risk averse agents

What I Expect You To Know

 How to write down participation and incentive compatibility constraints for both the principal and the agent

 Be able to do this in varying contexts

 Be able to draw a graph of when these bind

 Be able to differentiate adverse selection from moral hazard

-- end of part 5 --

Asymmetric Information: Applications

6. Auctions

Economics 313

History

 The word auction comes from the Latin for increase (related to augment)

 Very old method of selling.

 Customary to mention Herodotus reference to auctions. He was talking about Babylonian bride “auctions”

Wide usage  Auctions widely used both for buying and

selling, perhaps the most significant form of market transaction

 Art, houses, commodities, Treasury bills

 Typically think first of “open outcry” ascending bid auction

 Many complications have been studied; we’ll focus on single item auctions.

Four basic forms

 English: ascending bids

 Dutch: descending bids

 First-price sealed bid

 Second-price sealed bid (Vickery)

Strategic equivalents

Dutch First price

English Second price

<=>

<≈>

Performance

Revenue: taken from seller’s perspective

Efficiency: highest value wins  Resale is possible if not

There are sometimes concerns about ex post market structures

Valuations

1. (Independent) Private values

2. Common (interdependent) values

 Attitude toward risk is also important for some auction types

The benchmark model  Single good

 𝑛𝑛 risk neutral bidders, indexed by 𝑖𝑖 with values 𝑣𝑣𝑖𝑖

 𝑖𝑖𝑖𝑖𝑖𝑖 private values: bidders get signal of own value, and know the distribution of values.

 No budget constraint: i.e. can pay value.

First price auction

 Bid 𝑏𝑏𝑖𝑖  Payoffs

 If tie, toss a coin.

 To maximize expected utility, it is clear you must “shade your bid”:𝑏𝑏𝑖𝑖 < 𝑣𝑣𝑖𝑖

  

< >−

=Π ≠

≠

jiji

jijiii i bb

bbbv max if 0 max if

Second price auction  Bid 𝑏𝑏𝑖𝑖  Payoffs

 If tie, toss a coin.

 What is the optimal strategy for this auction?

  

< >−

=Π ≠

≠≠

jiji

jijijiji i bb

bbbv max if 0 max if max

Second price auction Weakly dominant strategy: bid valuation

iv Payoff:

j ij

b ≠

max ib

Second price auction Weakly dominant strategy: bid valuation

iv ib

j ij

b ≠

max Payoff:

0max <− ≠

j ij

i bv

Second price auction Weakly dominant strategy: bid valuation

iv Payoff:

* ib

}0,maxmax{ j ij

i bv ≠

−

Vickrey-Clarke-Groves mechanism  The previous result a special case of a

more general mechanism

 Intuition: make each agent bear the entire cost of misallocations caused by their misreports of valuations

 Think of bid as report of value

 If bid low or high, the total value to other participants – bidders and seller – is unchanged

VCG mechanism: bid too high

iv ib

j ij

v ≠

max

iv ib

j ij

v ≠

max

Allocation inefficiency = Loss to bidder j = gain to seller = loss to bidder i

VCG mechanism: bid too high

ib j

ij v

≠ max

Allocation inefficiency = loss to bidder i

Loss to seller=gain to bidder j

VCG mechanism: bid too low

Auction revenue in benchmark model  Expected revenue depends on the number of

bidders and their strategies  Can be computed with “order statistics”:

Random variables based on repeat draws from a distribution: e.g. median

 Rank the outcomes: what is the expected highest?

 The first order statistic is a random variable the realization of which is the highest draw (or sometimes lowest).

 Second order statistic is the second highest, etc.

Expected payment in second price auction  Win when highest bid: this is true when all the other

(independent) draws are less than yours. If you draw 𝑣𝑣 this happens with probability (𝐹𝐹 𝑣𝑣 )𝑛𝑛−1

 Pay FOS of remaining 𝑛𝑛 − 1 bidders, which is the SOS conditional on 𝑣𝑣 being the highest value

First price auction

 Highest bidder wins, pays own bid

 CLAIM: the optimal bid is to focus on the outcome where you have the highest valuation (otherwise you lose, and get 0), and then bid the FOS of the remaining 𝑛𝑛 − 1 bidders (conditional on your value being the highest).

 This is the optimal amount of “shading” of your bid.

Revenue equivalence

Suppose values are independently and identically distributed and all bidders are risk neutral.

Any symmetric and increasing equilibrium of any standard auction where the expected payoff of a bidder with a zero value is zero will produce the same expected revenue to the seller.

Other assumptions: Common value

 Sometimes “interdependent” values

 Problems is the so-called Winner’s curse

 Consider auctioning a jar of coins: the person who “guesses” highest will win, but if on average guesses are correct, this will be too high.

 With interdependent values optimal strategies condition on winning only in “optimistic” states.

Extensions

 Risk aversion: first price yields more revenue as optimal shading falls due to riskiness

 Reserve bids: trade off between revenue and efficiency. Generally increase revenue.

 Asymmetric value distributions: dominant bidders  Revenue equivalence fails

Shortcomings of the Vickrey Mechanism

 Combinatorial auctions: computational costly to calculate values for various combinations

 Reveal information: dynamic consequences

 Budget constraints: best strategy may depend on what others are bidding.

 Monotonicity problems: more bidders may lead to lower revenue!

100

Monotonicity problem  Example from Spectrum Actions  Two new entrant bidders, valuations

 Outcome: bidder gets both licenses for $900 𝑀𝑀.

pairfor 1$1 Bv = pairfor 900$2 Mv =

101

Monotonicity problem  Now include two incumbents, each wanting

one license only with valuations

 Outcome: incumbents each win one license for a total social value of

 But price is $0!

B1$

B2$

102

Monotonicity problem  To see why, recall that the price bidder

three, say, pays under the Vickery mechanism is the opportunity cost to the other bidders when bidder 3 wins.

 This is the maximum value for the two licenses together ($1B) minus the value for one license alone (also $1B)

 The opportunity cost is thus $0

103

Summary  Auctions are useful to allocate goods when

values are uncertain

 The VCG mechanism can lead to bidders revealing their valuations.

 Much work has been done adapting these insights to the design of auctions.

What I Expect You To Know

104

 What are the four basic forms of auction

 How the Vickery second price auction incentivizes bidders to bid their valuation

 The shortcomings of the Vickery mechanism.

-- end of part 6 --