eco question

DEMAND FOR INSURANCE

Why buy insurance?

 Demand for insurance driven by the fear of the unknown  Hedge against risk -- the possibility of bad outcomes

 Purchasing insurance means forfeiting income in good times to get money in bad times  If bad outcome does not occur, money spent on insurance is lost

 Ex: a person who buys health insurance but never visits the hospital might have been better off spending that income elsewhere

 A very simplified view is that insurance is like placing a bet with the insurance company that pays off if a bad outcome occurs

Utility and Risk Aversion  Risk aversion drives demand for insurance  How can we represent risk aversion using utility functions?

 Graphically,  Utility increasing with income: slope is positive

 Marginal utility decreasing: slope decreases as I increases

Adding uncertainty to the model

 An individual does not know whether she will become sick, but she knows the probability of sickness is p between 0 and 1  Probability of sickness is p

 Probability of staying healthy is 1 - p

 If she gets sick, medical bills and missed work will reduce her income  IS = income if she does get sick

 IH > IS = income if she remains healthy

Expected value

 The expected value of a random variable X, E[X], is the sum of all the possible outcomes of X weighted by each outcome’s probability of occurring  If the outcomes are x1, x2, . . . , xn, and the probabilities for each outcome

are p1, p2, . . . , pn respectively, then:

 E[X] = p1 x1 + p2 x2 + · · · + pn xn

 In the simple case from the previous slide there are only two outcome (healthy or sick), so the expected value of income E[I] is:

 E[I] = p IS + (1- p) IH

Example: expected value

 Suppose you are offered a choice between two possible options, a lottery and a certain payout:  A: a lottery that awards $500 with probability 0.5 and $0 with probability

0.5.

 B: a check for $250 with probability 1.

Example: expected value

 Suppose you are offered a choice between two possible options, a lottery and a certain payout :  A: a lottery that awards $500 with probability 0.5 and $0 with probability

0.5.

 B: a check for $250 with probability 1.

 The expected value of both the lottery and the certain payout is $250:

 E[I] = p IS + (1- p) IH  E[A] = .5(500) + .5(0) = $250

 E[B] = 1(250) = $250

People prefer certain outcomes

 Studies find that most people prefer certain payouts over uncertain scenarios

 If someone prefers the uncertain option (option A), what does that imply about their utility function?

 To answer this question, we need to define expected utility for a lottery or uncertain outcome.  That is, how do people make decisions to maximize utility

when at least one of the outcomes is uncertain?

Expected Utility Theory  The expected utility of a random payout X, denoted E[U(X)], is

the sum of the utility from each of the possible outcomes, weighted by each outcome’s probability of occurring.

 If the outcomes are x1, x2, . . . , xn, and the probabilities for each outcome are p1, p2, . . . , pn respectively, then:  E[U(X)] = p1 U(x1) + p2 U(x2) + · · · + pn U(xn)  Instead of calculating the expected value of X, we are

calculating the expected value of U (which is the expected level of utility).

 Expected utility theory simply says that when outcomes are uncertain, people maximize what they expect their utility to be

Example

 Preference for option B over option A implies that expected utility from B, is greater than expected utility from A:  E[U(B)] ≥ E[U(A)]

U($250) ≥ 0.5 U($500) + 0.5 U($0)

 In this case, even though the expected values of both options are equal, the certain payout is preferred over the less certain one  These preferences lead a consumer to act in a risk-averse manner

Expected utility without insurance

 Lottery scenario similar to case of insurance customer  Earn high income IH if healthy, and low income IS if sick.

 Uncertainty about which outcome will happen, though probability of becoming sick is known: ‘p’  Expected utility E[U(I)] is:

 E[U(I)] = p U(IS) + (1- p) U(IH)

E[U(I)] and probability of sickness

 Consider a case where the person is sick with certainty (p = 1):

 E[U] = U(IS) equals the utility from certain income IS (Point S)

 Consider case where person has no chance of becoming sick (p = 0):

 E[U] = U(IH) equals utility from certain income IH (Point H)

What if p lies between 0 and 1?

 For p between 0 and 1, expected utility E[U(I)] falls on a line segment between S and H

Example: p = 0.25  For p = 0.25, expected income is:

E[I] = 0.25·IS + (1 - .25)·IH  Expected utility when p=0.25 is

E[U(I)]0.25 = 0.25·U(IS) + (1 - 0.25)·U(IH)

 This outcome is depicted by Point A below, expected income is on the horizontal axis, expected utility is on the vertical axis

Example: p = 0.75  For p = 0.75, person’s expected income is:

E[I] = 0.75·IS + (1 - .75)·IH  Expected utility when p=0.75 is

E[U(I)]0.75 = 0.75·U(IS) + (1 - 0.75)·U(IH)

 This outcome is depicted by Point B below

Expected utility and expected income

 Important! Expected utility is not the same as utility from expected income (because the utility function is nonlinear)  Expected utility E[U(I)]

 Utility from expected income U(E[I])

 For risk-averse people, U(E[I]) > E[U(I)]

Definition of Risk Aversion Equivalent definitions of risk-aversion: A risk averse person  …prefers certain outcomes to uncertain outcome that

yield the same expected income

 …prefers the utility from expected income to the expected utility from uncertain income (holding fixed E[I])  U(E[I]) > E[U(I)]

 …has a strictly concave utility function with respect to income

1. Utility is increasing as income increases

2. Diminishing marginal utility of income

A basic health insurance contract

 Customer pays an upfront fee  Payment r is known as the insurance premium

 If ill, customer receives q, the insurance payout  If healthy, customer receives nothing

 Either way, customer loses the upfront fee  Customer’s final income is:

 Sick: IS + q – r

 Healthy: IH + 0 – r

Income with insurance

 Let IH’ and IS’ be income with insurance  Sick: IS’ = IS + q – r

 Healthy: IH’ = IH + 0 – r

 Remember that risk-averse consumers want to avoid uncertainty  For them, optimally IH’ = IS’ so income is the same regardless of being

healthy or sick  This is called “full insurance”

Full insurance payout

 Full insurance implies that income is state- independent (ie. Income does not depend on whether you are healthy or sick)

 IH’ = IS’  So

 IH + 0 – r = IS + q – r  IH = IS + q  q = IH – IS

 The payout from a full insurance contract is difference between incomes without insurance

Actuarially fair insurance

 Actuarially fair means that insurance is a fair bet  i.e. the premium equals the expected payout

r = p*q

 Insurer makes zero profit/loss from actuarially fair insurance in expectation

Actuarially fair, full insurance

 Why?  Step 1: actuarially fair means r=pq  Step 2: full insurance implies q=(IH-IS)

 Notice that consumers with actuarially fair, full insurance achieve the same expected income regardless of whether they are sick or healthy!

 If risk-averse individuals can purchase actuarially fair insurance they will always choose full insurance

Insurance and risk aversion

 As we have seen, simply by reducing uncertainty, insurance can make this risk-averse individual better off.

 Relative to having no insurance, with insurance she loses income in the healthy state and gains income in the sick state.  In other words, the risk-averse individual willingly sacrifices some good

times in the healthy state to ease the bad times in the sick state.

Insurer profits

 Now consider the same insurance contract from the point of view of the insurer  Premium r

 Payout q

 Probability of sickness p

 E[Π] = Expected profits

Fair and unfair insurance

 In a perfectly competitive insurance market with perfect information, profits will equal zero

 Same definition as actuarially fair!  An insurance contract which yields positive

profits is called actuarially unfair insurance:

 An insurer would never offer a contract with negative profits

Full vs. partial insurance

 With partial insurance, income still depends on whether the person ends up being healthy or sick

 Size of the payout q determines the fullness of the contract  The closer q is to IH – IS , the closer the insurance contract is to full

insurance

Comparing insurance contracts

 For anyone risk-averse, actuarially fair & full insurance contract offers the most utility

Comparing fair insurance contracts

 AF -- Actuarially fair & full  AP -- Actuarially fair & partial  A -- Uninsurance

 U(AF) > U(AP) > U(A)

Comparing unfair contracts

 AF – Full but actuarially unfair contract  AP – Partial but actuarially fair contract

Comparing unfair contracts

 In the previous graph, U(AF) > U(AP)

 Even though AF is actuarially unfair, its relative fullness (i.e. higher payout) makes it more desirable

 But notice if contract AF became more unfair, then expected income E[I] falls

 If too unfair, AF may generate less utility than AP

 Similarly, AP may become more full by increasing its payout

 Uncertainty falls, so point AP moves

 At some point, this consumer will be indifferent between the two contracts

 Where is that point?

Comparing unfair contracts

 AF’ – Full but actuarially unfair contract

 AP – Partial but actuarially fair contract

 In this case, both contracts give same expected utility

 In addition to the expected loss, how much extra money would this consumer be willing to pay for insurance?

 This amount is called the “risk premium,” and it describes the willingness to pay, or individual demand, for insurance

B

D

A

C

32• Willingness to pay for insurance is greater when: • There is larger variance in the potential wealth outcomes

• Eg Variance(Y1,Y2) < Variance (Y1,Y3) causes CD<AB • There is more uncertainty about which level of Y will occur

• Eg A 50/50 chance of high/low wealth is the largest amount of uncertainty. • If the probability of losing money is 99% the risk premium is LOWER (all else

equal) because the outcome is more certain

Utility

Wealth ($) Y3 Y2 E[Y] Y1

Willingness to Pay for Insurance

Supply of Insurance: Why are firms willing to sell insurance?

 Example: A Very Simple Insurance Pool:  Imagine 10 people each facing a risk of losing $100 with a

probability of 0.1  E(loss) = 0.1($100) + 0.9($0) = $10

 Each individual agrees to pay $10 and the ‘loser’ will get the $100  Actuarially fair premium is $10 (expected loss)

 Each individual ends up paying $10, and there is no remaining risk of losing money

 An insurance company is a firm that creates and manages an (or multiple) insurance pool(s)

 Simple idea is that consumers are averse to risk, but firms are risk- neutral, since they can diversify risk across many investors

33

Supply of Insurance: Why are firms willing to sell insurance?

 Insurance works by pooling risks across many people  Relies on the statistical Law of Large numbers

 As the number of pool members gets very large the average loss converges to the expected loss

 In this case, the insurance company’s revenue per customer (r) will equal is costs per customer (pq)

 This simple pooling of risk works if: 1. The probability of a loss can be predicted accurately 2. Losses are uncorrelated across people

 These two conditions do not always hold!!  Example: what would an insurance pool look like in the case of earthquake risk?  You should not be surprised to learn that in general a private, competitive

market for earthquake insurance will not function well (ie prices will likely be very actuarially unfair, and the market may not exist at all without regulation)

34