Pub Econ Lecture 12 Social Insurance

8/3/2019 Pub Econ Lecture 12 Social Insurance

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Public Finance

Dr. Katie Sauer

Social Insurance


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Figure 12-1: Government Spending, 1953 & 2007


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Expected Value

Expected value incorporates the probability of an

occurrence of an event with its outcome.

E[X] = p1X1 + p2X2 + + pnXn

pi is the probability of an event

Xi is the outcome associated with an event

The probabilities must always sum to 1.


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Suppose that in a year, there is a

0.1% chance that you will be involved in a caraccident resulting in $300,000 worth of damages

and injuries

10% chance that you will be involved in a caraccident resulting in $9000 worth of damages

What is the expected value of the damages?


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E[Damages] = (0.001)(300,000)

+ (0.10)(9,000)+ (0.899)(0)

E[Damages] = 300 + 900 + 0

= 1200

You should be willing to pay $1,200 per year for an

insurance policy that covers all of your expenses.

If your insurance company charges you $1,200, they are

charging an actuarially fair price.


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Actuarially fair pricing means your insurance firm willearn zero expected profit.

E[profit] = premium expected payout = 0

= premium (probability of payout)(payout amount)

Ex: E[profit] = 1200 (0.001)(300,000) (0.10)(9000)

= 1200 300 900= 0


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Ex: You are playing roulette.

- 38 pockets- choose red 16 pocket to bet $1 on

- if the ball lands in red 16, win $35

- if the ball lands in any other pocket, lose $1

Calculate your expected value of a $1 bet.

E[$1 bet on red 16] =

= (1/38)(35) + (37/38)(-1)= 0.92 0.97

= - 0.05


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Ex: Insurance

Sams income is $50,000 per year.

There is a 1% chance that Sam will get hit by a car next

year, resulting in $20,000 in medical expenses.

Sams utility function is U = C0.5 .

Sam can buy insurance form cents per dollar of

insurance coverage $b.

Assume an actuarially fair policy.


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Sams expected utility function can be expressed as:

E[U]=

(0.99)(50,000 mb)0.5 + (0.01)(50,000 mb 20,000+b)0.5

An actuarially fair policy means that the firm earns zero

expected profit:

E[] = mb (0.01)b = 0

m = 0.01


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Substituting in form yields:

E[U]

=(0.99)(50,0000.01b)0.5 + (0.01)(30,000 0.01b +b)0.5

Maximizing with respect to the level of insurance coverage

yields:

5.05.0 )99.0000,30(

)00495.0(

)01.0000,50(

)00495.0(

bbb

EU

!x

x


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Set it equal to zero and solve forb.

0)99.0000,30(

)00495.0(

)01.0000,50(

)00495.0(5.05.0

!

bb

bb 01.0000,5099.0000,30 !

000,20!b

It is optimal for Sam to buy $20,000 of coverage.

He will fully insure against the risk.


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If Sam wants $20,000 of coverage, he pays a premium of:

mb

m20,000

from earlier we found m = 0.01

(0.01)(20,000)

$200


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Calculate Sams Expected Utility:

- found m and b

E[U]=

(0.99)(50,000 mb)0.5 + (0.01)(50,000 mb 20,000+b)0.5

E[U]=

(0.99)(50,000200)0.5 + (0.01)(50,000 20020,000+

20,000)0.5

E[U]= 220.93 + 2.23 = 223.16


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What if Sam had only partially insured?

- paid $100 for $10,000 of coverage

E[U]=(0.99)(50,000100)0.5 + (0.01)(50,000 10020,000+

10,000)0.5

E[U]= 221.15 + 1.99 = 223.14


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The Expected Utility Model

Your utility takes the following form: U = C0.5

You are hit by a car with probabilityp.

If you get hit, your medical costs are .

Your income is W, regardless of whether you get hit.

You can buy insurance for a premium ofmper dollar of

insurance coverage.

Insurance will pay you $b if you are hit.

Insurance is actuarially fair.


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Your expected utility can be written as:

E[U] = (probability of not getting hit)(utility)

+ (probability of getting hit)(utility)

E[U] = (1 p) (W mb)0.5 + (p)(W mb + b)0.5


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Actuarially fair policy:

E[] = mbpb = 0

mb is the amount they receive in premiumspb is the expected payout


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mbpb = 0

Simplifying yields:

mb = pb

m = p

For example, if the risk of payout is 10%, then the

insurance firm will charge 10 cents per dollar ofcoverage.


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Substitute m = p into the expected utility.

E[U] = (1 p) (W mb)0.5 + (p) (W mb + b)0.5

E[U] = (1 p) (W pb)0.5 + (p) (W pb + b)0.5


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Now, maximize the expected utility by choice of

coverage (b).

5.05.0 )(

)5.0)(1(

)(

)5.0)(1(

bpbW

pp

pbW

pp

b

EU

!

x

x

H


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Set equal to zero and solve forb.

0)(

)5.0)(1(

)(

)5.0)(1(5.05.0

!

bpbW

pp

pbW

pp

H

5.05.0 )(

)5.0)(1(

)(

)5.0)(1(

bpbW

pp

pbW

pp

!

H

5.05.0 )()( pbWbpbW ! H


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? A ? A25.025.0

)()( pbWbpbW! H

pbWbpbW ! H

H!b

The optimal amount of coverage to buy is the amountthat exactly offsets the medical costs.

- full insurance


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Expected Utility Theory tells us that with actuarially fair

pricing, individuals will want to fully insure themselves

to equalize consumption in all states of the world.

People differ in their taste for risk.


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Given that it is theoretically optimal to have full

insurance, why have Social Insurance?

Asymmetric Information &Adverse Selection


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Suppose there are 100 careless insurance consumers

who dont pay attention when crossing the street.

5% chance of getting hit by a car$30,000 medical bills

Suppose there are 100 careful insurance customers

who always look both ways.

0.5% chance of getting hit by a car

$30,000 medical bills

The insurance firm doesnt know who is careless and

who is careful.


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Strategy 1: Ask each person if they are careful or not.

Everyone will say they are careful.

actuarially fair premium = (0.005)(30,000)

= $150

Total premiums = 200 x 150 = 30,000

E[profit] = 30,000 (0.005)(30,000)(100)

(0.05)(30,000)(100)

= 30,000 15,000 150,000= 30,000 165,000

The firm will lose money, so will not offer any insurance.

- market failure


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Strategy 2: Offer a pooled rate.

Divide the expected payout by the number of

people and charge that amount to everyone

165,000 / 200

= $825

Now, the careful consumers may choose not

to purchase the insurance.

Collect premiums from the careless:

100 x $825

= $82,500


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Pay out:

(0.05)(100)($30,000)

= $150,000

The insurance firm will still lose money.

The careful consumers will not be able to

purchase their optimal amount of insurance.- market failure


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OvercomingAsymmetric Information

1. Risk Premiums

Many people are risk averse and are willing to pay more

than the actuarially fair price for insurance.

If our careful consumers are risk averse, they may

be willing to pay $825 - $150 = $675 in a risk

premium.

Wed then have a pooling equilibrium.

- all fully insure, which is optimal

- not priced fairly for all


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2. Separate Products at Separate Prices

- try to get customers to reveal information

Option 1: $30,000 of coverage for $1,500

(0.05)(30,000) = 1,500

Option 2: $10,000 of coverage for $50

(0.005)(10,000) = 50

Likely that the careless would choose option 1 and the

careful would choose option 2.

This is called a separating equilibrium.

- not optimal for the careful (market failure)


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Government Involvement Rationale:

- asymmetric information / adverse selection

- externalities

- uninsured motorists

- administrative costs

- economies of scale

- redistribution- tax low-risk to subsidize high-risk

- paternalism


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Social Insurance vs Self-Insurance

Suppose you are unemployed. Forms of self-insurance to

get you through:

- personal savings

- borrow- money from other members of household

- money from extended family, friends, church

Suppose you can receive Unemployment Insurance.The UI Replacement Rate is the ratio of UI

benefits to pre-unemployment earnings.


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UI replacement rate and the drop in consumption

(a) No savings, credit cards, money from friends

- at 0% UI replacement rate, consumption falls 100%



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(b) Some savings, credit cards, money from friends



- use UI instead of private funds


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(c) full private source of funds



- use UI instead of private funds


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The Consumption-Smoothing Role of Social Insurance

When events are predictable, social insurance plays a

smaller consumption-smoothing role.

When events are less costly, social insurance plays a

smaller consumption-smoothing role.


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Optimal Social Insurance Systems:

should partially, not completely insure people against

adverse events.

Benefits: consumption smoothing

Costs: moral hazard

Pub Econ Lecture 12 Social Insurance

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