Economics of Uncertainty and Insurance · Economics of Uncertainty and Insurance Hisahiro Naito University of Tsukuba January 11th, 2013 ... Assume that consumer is risk-averse which

Economics of Uncertainty and Insurance

Hisahiro Naito

University of Tsukuba

January 11th, 2013

Hisahiro Naito (University of Tsukuba) Economics of Uncertainty and Insurance January 11th, 2013 1 / 31

Introduction

This section talks about the economics under uncertainty andinsurance

First, it introduces the expected utility theory and then study thedemand for insurance


Expected Utility Theory

People buy insurance to prepare for the evens that does not happenwith 100 percent probability.

Examples are tra¢ c accidence, health insurance for traveling inforeign countries, insurance for �re etc.

We need to study the behavior under uncertainty.


Expected Utility Theory (2)

von-Neumann and Oscar Morgenstein show that the human behaviorunder uncertainty can be explained as maximization of the expectedutility maximization.

Not that expected utility maximization is di¤erent from maximizationof the utility of expected value.



Consider the following examples.

Let the von-Nueman Morgenstein utility function be u(c)(NM utilityfunction)

Assume that person�s income become x1 with probability p1 and x2with probability 1� p1.The expected value of income is p1x1 + (1� p1)x2The utility from the expected value is u(p1x1 + (1� p1)x2)The expected utility is p1u(x1) + (1� p1)u(x2)



NM utility function u(c) is called risk-averse when u(c) is concave

NM utility function is risk lover when u(c) is convex

NM utility function is risk-neutral when u(c) is linear.



To see why, consider the following event.

Assume that with probability of 0.5, this person has income 90. Withprobability of 0.5, this person has income, 110.

The expected value of income is 100.

How about the expected utility?



MN function is concave

9

90 100 110



NM utility function is convex

9

90 100 110



NM function is linear

9

90 100 110



Consider the following examples. Assume that u(c) = log c

Then, ln(100)=4.605. 0.5ln(90)+0.5ln(110)=4.600.

Thus, this consumer prefer certain event with the same expectedvalue.

On the other hand, assume that u(c) = c2

Then, 1002 = 10000. 0.5� 902 + 0.5� 1102 = 10100. Thus, thisconsumer prefers uncertain events with the same expected values.



In general, it is believed that people are risk-averter. Thus, it isreasonable to assume that NM utiltiy function is concave.

In general, expected utility is written as

∑all s

p(s)(xs )

where p(s) is the probability that even s happens

xs is the income when event s happens


Law of Large Number

Now consider pooling of income risk.

Suppose that there are two person. Person i has income xi which israndom variable. The expected value of xi ,E (xi ) = µ.

The variance of xi is σ2.

Also assume that the �uctuation of person A�s income is independentof person B�s income.

In other words, cov(x1, x2) = 0.


Law of Large Number

Now, suppose that person A and person B agree that they sum theirincome and divided it by them.

This implies that each person income becomes xA+xB2

The expected value of xA+xB2 is E ( xA2 ) + E (xB2 ) =

µ2 +

µ2 = µ

The variance ofxA+xB2 = var( xA2 +

xB2 ) =

14var(xA) +

14var(xB ) +

12cov(xA,xB ) =

12σ2

The expected value is the same and the variance becomes smaller.

When there are N person and N persons�s income is independent.Then the variance of shared income is

σ2

N

Thus, the variance become smaller as N increases. This property iscalled the Law of Large Number.


Insurance Demand

Law of large number is the motive of the supply side of insurance

How about the demand side?

Consider a case that an individual initially has $100. With probabilityp1 some accident happen. If accident happens, his income x1 which islower than 100. . With probability 1� p1, his income is still 100.Suppose that there is an insurance. The price of insurance(calledpremium) is pr .

If a person purchase one unit of this insurance, he get $1 when he hasan accident.

Let y be the amount of insurance that he purchases.

How much does he purchase?


Insurance Demand

The consumer will maximize the following utility function and chooseoptimal y :

p1u(x1 � pr � y + y) + (1� p1)u(100� pr � y)

Now consider the expected pro�t of insurance company

Expected pro�t of selling one unit of insurance is

pr � p1 � 1+ (1� p1)� 0If free entry is allowed for insurance company, the expected pro�tshould be equal to zero in the long run.

Thus, pr = p1 � 1This is called actuarially fair premium. In other words, actuarially fairpremium is the premium that will make the expected pro�t ofinsurance company equal to zero.


Insurance Demand

The optimal amount of insurance purchase must satisfy the followingFOC:

p1u0(x1 � pr � y + y)� (1� pr) + (1� p1)u(100� pr � y)� (�pr) = 0orp1u0(x1 � pr � y + y)� (1� pr) = (1� p1)u(100� pr � y)� pr

When the insurance premium is fair, pr = p1. Thus

p1u0(x1 + (1� p1)� y)� (1� p1) = (1� p1)u(100� p1 � y)� p1u0(x1 + (1� p1)� y) = u(100� p1 � y)

x1 + (1� p1)� y = 100� p1 � yy = 100� x1


Insurance Demand

Thus, this individual purchases the insurance 100� x1 units.With this insurance, when the accident happens, his income isx1 � (100� x1)pr + 100� x1 = 100� p1(100� x1)Without the accident, his income is100� pr � (100� x1) = 100� p1 � (100� x1)Thus, this consumer will choose the full insurance.

This holds in general

As long as insurance premium is fair and consumers are risk averse,consumers will choose the full insurance so that the consumption indi¤erent events become the same


State Dependent Consumption

It is useful to analyze the demand of insurance in the state dependentconsumption model

Expected utility model nicely �ts into the state dependentconsumption

In the economy, there are two state of nature

Accident state and non-accidnet state.

Consumption is contingent on those two state


State Dependent consumption(2)

Accident state is denoted as subscript a and non-accident is denotedas subscript n.

Consumption of accident state is denoted as ca.

p(a) is the probability that the accident happens.

1� p(a) is the probability that accident does not happenu(c) is NM utility function.

Assume that consumer is risk-averse which implies that u(c) isconcave


State dependent consumption

Expected utility is de�ned as

p(a)u(ca) + (1� p(a))u(cn)

Because u(c) is concave, the indi¤erence curve of (ca, cn) is convexto the origin.

It is useful to calculate the MRS at the 45 degree line.

Measure the cn on the horizontal line. Measure ca on the vertical line

Then MRS is(1� p(a))u0(cn)p(a))u0(ca)

At the 45 degree line, cn = ca. The slope of MRS=(1-p(a))/p(a)


Budget set

Let x be the initial income. D is the damage in the case of accident.Let pr be the premium of the one unit of insurance. y be the amountof insurance

cn = x � pr � yca = x �D � pr � y + y= x �D + (1� pr)y

ca = x �D + (1� pr)(x � cn)1pr

Assume that the free entry of insurance market. In this case,pr = p(a)The expected pro�t of insurance company is equal to zero

pr � y + p(a) � y = 0p(a) = pr

Thus, the budget constraint becomes

ca = x �D + (1� p(a))(x � cn)1p(a)


Budget set (2)

The budget set pass through (x-D, x)

The absolute value of the slope is (1-p(a))/p(a)


Expected Utility maximization

From the graph, the utility maximization is achieved at the 45 degreeline

cn = caFull insurance is achieved


Expected Utility maximization

From the graph, the utility maximization is achieved at the 45 degreeline

cn = caFull insurance is achieved


Problems in Insurance Market

Generally, there are two types of problems in the insurance market

This is due to the asymmetric information between insurer and insuree


Moral Hazard

Often, insurance market, insurer does not completly observe thebehavior of the insuree

This unobservability cause the so called moral hazard in insurancemarket

Due to the moral hazard problem, private insurance companiesintroduce some mechanism to reduce the moral hazard behavior.


Adverse selection

Consider an insurance market where there are several times ofconsumers: high-risk, middle-risk, low-risk.

The probablity that accident happens is relatively low to low riskconsumers.

The probability that accident happens is relatively high to high riskconsumer.

Middle risk consumer is between two types


Adverse Selection (2)

Suppose that initially three types of consumer joins the insurance andinsurance company cannot distinguish three types.

Then, insurance company will change the premium based on theaverage of those three groups.

However, such an insurance is not attractive for low risk people.

Thus, low risk consumer will opt out.


Adverse Selection (3)

Then, insurance premium will go up.

Then, the premium is not attractive even for middle risk people.

Thus, the middle risk consumer will opt out.

As a result, only the high risk consumers remain in the market.

Other consumers will opt out and the market for those consumers donot exit.

This is called adverse selection


Public Policy Implication of Moral Hazard and AdveseSelection

Adverse Selection justi�es the public intervention of the insurancemarket

Why?

Moral hazard problem suggests that full insurance is the best policy

Why?


Documents

Economics of Uncertainty and Insurance · Economics of Uncertainty and Insurance Hisahiro Naito University of Tsukuba January 11th, 2013 ... Assume that consumer is risk-averse which