CHAPTER 7 DEMAND FOR INSURANCE Econ3004/ Econ6039 Health Economics, 2023 Semester 2 Dr Yijuan Chen, Australian National University Bhattacharya, Hyde and Tu – Health Economics 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 times avoided, then money lost

 Ex: The individual who buys health insurance but

never visits the hospital might have been better off

spending that income elsewhere.

Bhattacharya, Hyde and Tu – Health Economics Risk aversion

 Hence, risk aversion drives demand for

insurance

 We can model risk aversion through utility from

income U(I) Utility increases with income: U(I) > 0 Marginal utility for income is declining: U(I) < 0 Bhattacharya, Hyde and Tu – Health Economics Income and utility  Graphically, Utility increasing with income

U’(I) > 0 Marginal utility decreasing U’’(I) > 0 Bhattacharya, Hyde and Tu – Health Economics 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

Bhattacharya, Hyde and Tu – Health Economics 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

 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 our individual’s case, the formula for expected

value of income E[I]:

E[I] = p IS + (1- p) IH Bhattacharya, Hyde and Tu – Health Economics Example: expected value  Suppose we offer a starving graduate student 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 Bhattacharya, Hyde and Tu – Health Economics People prefer certain outcomes  Studies find that most people prefer certain

payouts over uncertain scenarios  If a student says he prefers uncertain option,

what does that imply about his utility function?

 To answer this question, we need to define

expected utility for a lottery or uncertain

outcome.

Bhattacharya, Hyde and Tu – Health Economics Expected Utility  The expected utility from a random payout X

E[U(X)] is the sum of the utility from each of the

possible outcomes, weighted by each outcome’s

probability.

 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)

Bhattacharya, Hyde and Tu – Health Economics Example  The student’s preference for option B over option A

implies that his expected utility from B, is greater

than his 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 student prefers the

certain payout over the less certain one.  This student is acting in a risk-averse manner over the

choices available.

Bhattacharya, Hyde and Tu – Health Economics Expected utility without insurance  Lottery scenario similar to case of insurance

customer  She gains a high income IH if healthy, and low

income IS if sick.

 Uncertainty about which outcome will happen,

though she knows the probability of becoming

sick is p  Expected utility E[U(I)] is:

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

Bhattacharya, Hyde and Tu – Health Economics  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) E[U(I)] and probability of sickness Bhattacharya, Hyde and Tu – Health Economics What if p lies between 0 and 1?  For p between 0 and 1, expected utility falls on a

line segment between S and H

Bhattacharya, Hyde and Tu – Health Economics Ex: p = 0.25  For p = 0.25, person’s expected income is:

E[I] = 0.25·IS + (1 - .25)·IH  Utility at that expected income is E[U(I)] (Point A) Bhattacharya, Hyde and Tu – Health Economics Expected utility and expected income  Crucial distinction between  Expected utility E[U(I)] Utility from expected income U(E[I]) For risk-averse people, U(E[I]) > E[U(I)] Bhattacharya, Hyde and Tu – Health Economics Risk-averse individuals

Synonymous definitions of risk-aversion:  Prefer certain outcomes to uncertain ones with the

same expected income.

 Prefers the utility from expected income to the

expected utility from uncertain income

 U(E[I]) > E[U(I)]  Concave utility function  U’(I) > 0  U’’(I) < 0

Bhattacharya, Hyde and Tu – Health Economics 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

Bhattacharya, Hyde and Tu – Health Economics 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’ Bhattacharya, Hyde and Tu – Health Economics Full insurance  Full insurance means full of certainty, i.e. no

income uncertainty IS’ = IH’  Final income is state-independent Regardless of healthy or sick, final income is the

same  Risk-averse individuals prefer full insurance to

partial insurance (given the same price) Bhattacharya, Hyde and Tu – Health Economics Full insurance payout  State independence implies 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

Bhattacharya, Hyde and Tu – Health Economics 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 Bhattacharya, Hyde and Tu – Health Economics Actuarially fair, full insurance Notice consumers with actuarially fair, full insurance achieve their expected income with certainty! Bhattacharya, Hyde and Tu – Health Economics Insurance and risk aversion  As we have seen, simply by reducing uncertainty,

insurance can make this risk-averse individual

better off.

 Relative to the state of no insurance, with

insurance she loses income in the healthy state

(IH > IH) and gains income in the sick state (IS <

IS).

 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.

Bhattacharya, Hyde and Tu – Health Economics 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[] = Expected profits Bhattacharya, Hyde and Tu – Health Economics Fair and unfair insurance  In a perfectly competitive insurance market, profits

will equal zero  Same definition as actuarially fair!  An insurance contract which yields positive profits is

called unfair insurance:  An insurer would never offer a contract with

negative profits Bhattacharya, Hyde and Tu – Health Economics Full vs. partial insurance  Partial insurance does not achieve state- independence  Size of the payout q determines the fullness of the

contract

 Closer q is to IH – IS , the fuller the contract Bhattacharya, Hyde and Tu – Health Economics Comparing insurance contracts  AF -- Actuarially fair & full  AP -- Actuarially fair & partial  A -- Uninsurance  U(AF) > U(AP) > U(A) Bhattacharya, Hyde and Tu – Health Economics The ideal insurance contract  For anyone risk-averse, actuarially fair & full

insurance contract offers the most utility  Hence, it is called the ideal insurance contract  Ideal and non-ideal insurance contracts: Bhattacharya, Hyde and Tu – Health Economics Comparing non-ideal contracts  AF – Full but actuarially unfair contract  AP – Partial but actuarially fair contract Bhattacharya, Hyde and Tu – Health Economics Comparing non-ideal contracts  In this case, 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

Bhattacharya, Hyde and Tu – Health Economics Conclusion  Demand for insurance driven by risk aversion  Desire to reduce uncertainty  Diminishing marginal utility from income  U(I) is concave, so U’’(I) < 0  U(E[I]) > E[U(I)]  Risk aversion can explain not only demand for

insurance but can also help explain  Large family sizes  Portfolio diversification  Farmers scattering their crops and land holdings

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