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.
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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
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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)
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 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
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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)
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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)]
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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’
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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)
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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
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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
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Actuarially fair, full insurance
Notice consumers with actuarially fair, full
insurance achieve their expected income with
certainty!
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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
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Comparing insurance contracts
 AF -- Actuarially fair & full
 AP -- Actuarially fair & partial
 A -- Uninsurance
 U(AF) > U(AP) > U(A)
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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:
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Comparing non-ideal contracts
 AF – Full but actuarially unfair contract
 AP – Partial but actuarially fair contract
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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