eco question

Health Economics Econ 5860 Prof. Kurt Lavetti

Rothschild-Stiglitz Model: Basic Setup

 Consider individuals who begin with wealth W  Face a risk of accident that will cause loss d  Can buy insurance with premium cost a1  Insurance pays back some money if accident occurs:

 Wealth if no accident: W-a1  Wealth with accident: W-d+a2

 Insurance contracts can be described by {a1, a2}

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Demand for Insurance

 Probability of accident is p  Consumers maximize their expected utility by picking the best

insurance contract {a1, a2} that is available in the market  They can always choose {0,0} which is no insurance

 Choose {a1, a2} to maximize p*U(W-d+a2)+(1-p)*U(W-a1)

 How can we represent these preferences over insurance contracts graphically…

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Wealth State Spaces 4

U

Wealth W-d W

Wealth if Sick

Wealth if Healthy

E W-d

W

Partial Insurance 5

U

Wealth

Wealth if Sick

Wealth if Healthy

W-d E

F W-d+a2

Question: What does the graph of full insurance look like?

Full Insurance 6

U

Wealth

Wealth if Sick

Wealth if Healthy

W-d E

F W-d+a2

All full insurance contracts are on the 45 degree line—if you are fully insured then wealth is the same if you are sick or healthy

45 Degree Line

Indifference Curves 7

Wealth if Sick

Wealth if Healthy

E

F G

IH

45 Degree Line

What can we say about preferences over these outcomes?

Indifference Curves 8

Wealth if Sick

Wealth if Healthy

E

F G

IH

45 Degree Line

Insurance Firm’s Problem

 In a competitive insurance market firms will earn zero profits

a1*(1-p)+a2*p=0 a2/a1 = -(1-p)/p

 a2 is how much the insurer pays the consumer if they get sick, and a1 is how much consumers pay to buy the insurance  In the firm’s profit function above, a1>0, a2<0  The ratio tells us the rate at which insurance allows the

consumer to trade wealth in the healthy state for wealth in the sick state

 -(1-p)/p is like an exchange rate between wealth in the two states

9

The Zero Profit Line 10

Wealth if Sick

Wealth if Healthy

E

F

G

I

Slope of dashed line= -(1-p)/p

What happens to insurer profits if they offer a contract that moves the consumer to point F? What about to point I?

The Zero Profit Line 11

Wealth if Sick

Wealth if Healthy

E

F

G

I

Zero Profit Line

What happens to insurer profits if they offer a contract that moves the consumer to point F? What about to point I?

Unprofitable Region

Profitable Region

The Zero Profit Line 12

Wealth if Sick

Wealth if Healthy

E

F

G

I

Zero Profit Line

Question: What happens to this figure if p is really large?

Unprofitable Region

Profitable Region

13

Wealth if Healthy

E Zero Profit Line

Unprofitable Region

Profitable Region

Wealth if Sick

The Zero Profit Line

A large value of p means the slope of the zero profit line, -(1-p)/p, gets closer to zero

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Wealth if Healthy

E

Zero Profit Line

Unprofitable Region

Profitable Region

Wealth if Sick

What happens to the zero profit line if p is low?

Unprofitable Region

The Zero Profit Line

Feasible Contracts 15

Wealth if Sick

Wealth if Healthy

E Zero Profit Line

Full Insurance Line

Indifference Curve

4

1

2

3

Question: Why will insurance contracts never exist in regions 1, 2, or 3?

Feasible Contracts 16

Wealth if Sick

Wealth if Healthy

E Zero Profit Line

Full Insurance Line

Indifference Curve

4

1

2

3

Region 4 is the set of potentially feasible insurance contracts— consumers are at least as well off in region 4 as they are at point E, and insurers don’t lose money

Equilibrium

 There are many feasible points, which (if any) is an equilibrium?

 First need to define what an equilibrium means in this model

 Definition of equilibrium 1. No equilibrium contract can make negative

expected profits 2. There is no contract outside of the equilibrium set

that could earn zero or positive profits

 We also still assume that consumers choose the contract from the equilibrium set that maximizes expected utility

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Case 1: Identical Consumers

18

Equilibrium with Identical Consumers

19

Wealth if Sick

Wealth if Healthy

Zero Profit Line

Full Insurance Line

• All individuals have the same probability p of getting sick

• Consider point F, which is within the set of potentially feasible contracts

• Is F an equilibrium contract?

E F

Equilibrium with Identical Consumers

20

Wealth if Sick

Wealth if Healthy

Zero Profit Line

Full Insurance Line

• To answer this, let’s verify whether F satisfies the 2 equilibrium condition

• Condition 1: F is below the zero profit line, so it satisfies condition 1

• Condition 2: Since F is not along the zero profit line, insurers will earn strictly positive profits. Therefore a new insurer could enter and offer contract G and steal all of the customers

• Therefore contract F cannot satisfy equilibrium condition 2

E F

G

Equilibrium with Identical Consumers

21

Wealth if Sick

Wealth if Healthy

Zero Profit Line

Full Insurance Line

• Repeating the same logic, only contracts along the zero profit line can satisfy condition 2

• With identical consumers there is only 1 equilibrium contract: point H

• At this point all consumers have full insurance, and all insurers earn zero profitsE

H

Equilibrium with Identical Consumers

22

Wealth if Sick

Wealth if Healthy

Zero Profit Line

Full Insurance Line

• Notice the importance of this conclusion!!

• A large number of insurers competing for customers

• Each insurer is free to sell any of the feasible contracts, potentially providing consumers with lots of choices

• Instead, we just showed this cannot possibly happen! Only one single insurance plan (H) can be sold in equilibrium

E

H

Case 2: Two Different Types of Consumers

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Consumer Types

 Suppose there are high risk and low risk types  High risk types have a probability of accident pH  Low risk types have a probability of accident pL  pH > pL  The fraction of the population that is high risk types is Z  Overall average probability of accident equals:

P=Z*pH+(1-Z)*pL  Consumers know their own types, but insurance companies

cannot observe types  Insurer only knows the probabilities and Z

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Equilibria with Two Types

 There are two types of equilibria that we need to consider:  Pooling equilibrium: both types buy the same insurance contract

 Separating equilibrium: high risk types buy different insurance contracts than low risk types

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Pooling Equlibrium 26

Wealth if Sick

Wealth if Healthy

Zero Average Profit Line

Full Insurance Line

• Question 1 : what is the slope of the zero profit line if we consider only pooling equilibria?

• Question 2: what will the indifference curves of the high risk types and low risk types look like?

E

Pooling Equlibrium 27

Wealth if Sick

Wealth if Healthy

Full Insurance Line

• High risk types are willing to give up more income in the healthy state in order to gain income in the sick state, because they know they are more likely to be sick

• Indifference curves of high risk types are flatter than indifference curves for low risk types

Zero Average Profit Line

E

UL

UH

F

Pooling Equlibrium 28

Wealth if Sick

Wealth if Healthy

Full Insurance Line

• Suppose F is a pooling equilibrium

• We know it must be on the zero average profit line in order to be an equilibrium

• Does F meet the second equilibrium condition?

Zero Average Profit Line

E

UL

UH

F

Pooling Equlibrium 29

Wealth if Sick

Wealth if Healthy

Full Insurance Line

• Consider a point G that is very very close to F, but lies between the indifference curves

• Low types prefer G to F • High types prefer F to G • The zero profit line for

high types is flatter than the zero profit line for low types, and flatter than the average zero profit line

• Therefore there must exist a point G that earns at least zero profits, and that low types prefer to F

• There cannot ever exist a pooling equilibrium

Zero Average Profit Line

E

UL

UH

F G

Can There be a Separating Equilibrium?

 In order for this to happen, it must be that insurers offer at least 2 different contracts

 High types all prefer one contract more than the others  Low types all prefer a different contract more than the others  Each of the preferred contracts earns at least zero profits  Now there will be two zero profit lines, one for low risk types

[slope = -(1-pL)/pL] and a different line for high risk types [slope = -(1-pH)/pH]

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Separating Equilibrium: Zero Profit Lines

31

Wealth if Sick

Wealth if Healthy

Average Zero Profit Line

Full Insurance Line

• The low risk zero profit line is steeper than the high risk zero profit line because pH > pL

E

Low Risk Zero Profit Line

High Risk Zero Profit Line

Separating Equilibrium: Symmetric Information

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Wealth if Sick

Wealth if Healthy

Full Insurance Line

• What would happen if insurers can tell which people are high risk and which are low risk?

• Insurers would offer contract G to low risk types and contract F to high risk types

• Low risk types would prefer contract G to contract F, but insurers would refuse to sell them contract GE

Low Risk Zero Profit Line

High Risk Zero Profit Line

F

G

UL

UH

Separating Equilibrium: Asymmetric Information

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Wealth if Sick

Wealth if Healthy

• Now return to the situation where insurers cannot tell who is high risk and who is low risk

• Insurers cannot offer contract G to anyone because high risk types will buy it and insurers will earn negative profits

E

UL

UH F

G

H

Separating Equilibrium: Asymmetric Information

34

Wealth if Sick

Wealth if Healthy

• Consider contract H with the following properties

• High risk types are indifferent between contract H and the perfect insurance contract F (so suppose they choose full insurance at F)

• Low risk types strictly prefer contract H to contract F

• This creates a separating equilibrium: high risk types all choose F and low risk types all choose H

• Does H satisfy all of the equilibrium criteria?

E

UL

UH F

G

H

Separating Equilibrium: Asymmetric Information

35

Wealth if Sick

Wealth if Healthy

• Does H satisfy all of the equilibrium criteria?

• All consumers choose the utility maximizing contract

• All contracts earn zero profits

• Can another insurer enter the market with a new contract and disrupt the equilibrium?

E

UL

UH F

G

H

Separating Equilibrium: Asymmetric Information

36

Wealth if Sick

Wealth if Healthy

• Suppose the new firm offers a contract at any point below the UL indifference curve

• This contract will only attract the high risk types, and the insurer will either lose money or consumers will prefer contract F and the new entrant will not sell anything

E

UL

UH F

G

H

Separating Equilibrium: Asymmetric Information

37

Wealth if Sick

Wealth if Healthy

• Suppose the new firm offers a contract at any point in region J

• Every consumer will prefer this contract over contracts H or F

• Everyone will switch to the new contract, creating a single pool

• The insurer will earn negative profits and go out of business

• Therefore no contract in region J can disrupt the separating equilibrium

• Anything above and to the right of zero profit line EG will also be unprofitable

E

UL

UH F

G

H

J

Separating Equilibrium: Asymmetric Information

38

Wealth if Sick

Wealth if Healthy

• Anything above or to the right of zero profit line EG will also be unprofitable

• Therefore there is no possible contract that can disrupt this equilibrium, so it satisfies the definition of an equilibrium

• Separating equilibria can exist

E

UL

UH F

G

H

J

Separating Equilibrium: Do they Always Exist?

39

Wealth if Sick

Wealth if Healthy

E

UL

UH F

G

H

Separating Equilibrium: Do they Always Exist?

40

Wealth if Sick

Wealth if Healthy

• Suppose the fraction of low risk types is very high

• Then the average zero profit line is steeper than it was in the last diagram, and intersects UL

• There is now a region K that didn’t exist before

• Can a new insurer enter and offer a contract in region K?

E

UL

UH F

G

H

K

Separating Equilibrium: Do they Always Exist?

41

Wealth if Sick

Wealth if Healthy

• Can a new insurer enter and offer a contract in region K?

• Yes, both high and low types will switch to any contract in region K, and that contract will earn non- negative profits

• However, since we know no pooling equilibrium can exist, any contract in region K will simply disrupt the separating equilibrium without creating a new equilibrium

• Therefore no equilibrium will exist in the long run

E

UL

UH F

G

H

K

Conclusions

 If information is symmetric  All consumers will have perfect insurance, and insurers will

discriminate between healthy and sick people, charging different prices for insurance

 If information is asymmetric  There cannot ever be a pooling equilibrium

 It is possible to achieve a separating equilibrium, but they do not always exist

 If a separating equilibrium does exist, high risk people will be fully insured, but low risk people cannot buy full insurance

 There may not be any equilibrium

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Welfare Implications

 If an equilibrium exists, some consumers will be unable to buy all of the insurance that they would like at the market price

 If people were willing to reveal their true risk types, everyone would be better off

 High risk people cause an externality by preventing low risk people from being able to buy full insurance  Low risk people would be better off if high risk people didn’t exist

 However high risk people gain nothing from this externality— they would be no better off if low risk people didn’t exist

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