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ChAPTER 7 Demand for Insurance

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: IH + q – r

Healthy: IS + 0 – r

Bhattacharya, Hyde and Tu – Health Economics

Income with insurance

Let IH’ and IS’ be income with insurance

Sick: IH’ = IH + q – r

Healthy: IS’ = IS + 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 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’

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

Bhattacharya, Hyde and Tu – Health Economics

184 | Modern Health Economics

q. Her incomes in the two states are thus:

Healthy: I0H = IH � r Sick: I0S = IS � r + q

(7.9)

Recall that the individual’s goal in buying insurance was to achieve an income of E[I]p with certainty, whether she is healthy or sick. What the individual would like most is:

E[I]p = I0H = I 0

S (7.10)

An insurance contract that fulfills Equation 7.10 is said to be actuarially fair, full insurance. We discuss these terms in more detail shortly.

Let us consider an insurance contract X with the following parameters. In this contract, assume the individual receives the difference between her healthy income and sick income if she is sick: q = IH � IS. In addition, assume that the premium is set such that the contract represents a fair bet r = pq. On average, the individual neither gains nor loses income from this contact.

The following algebra shows that with contract X, the individual’s income is E[I]p regardless of whether she turns out to be healthy or sick. In each column we start with Equation 7.9 and substitute in the parameters of this insurance contract. In the second line, we substitute r = pq, and in the third line, we substitute q = IH � IS.

⌅ Healthy State

I0H = IH � r = IH � pq = IH � p(IH � IS) = pIS + (1 � p)IH

I0H = E[I]p

⌅ Sick State

I0S = IS � r + q = IS � pq + q = IS � p(IH � IS) + (IH � IS) = pIS + (1 � p)IH

I0S = E[I]p

With this contract, the individual can receive E[I]p with certainty. This enables her to achieve points on the utility function, like A0, in Figure 7.3, whereas be- fore she was only able to achieve points on line segment H S below the utility- income curve. With the insurance contract, the individual’s utility increases even though her income does not. The insurance contract creates utility seemingly out of nowhere; simply by reducing uncertainty, the insurance contract can make the risk-averse individual better off.

Chapter 7: Demand for Insurance | 185

The nature of the insurance contract is that the individual loses income in the healthy state (IH > I0H ) and gains income in the sick state (IS < I

S) relative to the state of no insurance. This is the sense in which the insurance contract acts as an instrument that transfers income from the healthy state of the world to the sick state. The risk-averse individual willingly sacrifices some good times in the healthy state to ease the bad times in the sick state.

Fair and unfair insurance Consider now the same insurance contract we have been discussing from the point of view of the insurance company. Let E[⇧] be the expected profits that the insurer makes from offering a contract with premium r and payout q to any customer with probability of sickness p. If the customer actually stays healthy, the firm earns r dollars. On the other hand, if the customer falls ill, the firms still receives the premium r but loses the payout q. By applying the formula for expected value (see Equation 7.3), we find:

E[⇧(p, q, r)] = (1 � p)r + p(r � q) = r � pq

(7.11)

In a perfectly competitive insurance market, profits will equal zero. Just like in any competitive market, if profits were positive, new entrants would compete away those profits until all the firms left in the market would be making zero prof- its. If the profits were negative, then the insurer is giving money away to customers in the long run and will go out of business. Firms leave the market until profits reach zero. Setting expected profits to zero in Equation 7.11 implies r = pq. This condition is known as actuarial fairness.

Definition 7.1

Actuarially fair insurance contract: An insurance contract which yields zero profit in expectation; also called fair insurance.

E[⇧(p, q, r)] = 0 =) r = pq (7.12)

An insurance contract which yields positive profits is called unfair insurance.

E[⇧(p, q, r)] > 0 =) r > pq (7.13)

When insurance is fair, in a sense, it is also free. The customer’s expected in- come does not change from buying the contract, so she effectively pays nothing for it. Despite the fact that the premium r is positive in an actuarially fair contract,

Chapter 7: Demand for Insurance | 185

The nature of the insurance contract is that the individual loses income in the healthy state (IH > I0H ) and gains income in the sick state (IS < I

E[⇧(p, q, r)] = (1 � p)r + p(r � q) = r � pq

(7.11)

Definition 7.1

Actuarially fair insurance contract: An insurance contract which yields zero profit in expectation; also called fair insurance.

E[⇧(p, q, r)] = 0 =) r = pq (7.12)

An insurance contract which yields positive profits is called unfair insurance.

E[⇧(p, q, r)] > 0 =) r > pq (7.13)

Chapter 7: Demand for Insurance | 187

Definition 7.2

Full insurance contract: an insurance contract that achieves state indepen- dence; income in all states is equal.

I0S = I 0

Partial insurance contract: an insurance contract that is state-dependent; in- come in the sick state is still less than income in the healthy state.

I0S < I 0

Just as we derived the premium r in the cases of actuarially fair and unfair insurance, we can derive the payout q in the cases of full and partial insurance. We rely on the state-independence property of full insurance and the state-dependence property of partial insurance:

⌅ Full insurance

I0S = I 0

H IS � r + q = IH � r

IS + q = IH q = IH � IS

⌅ Partial insurance

I0S < I 0

H IS � r + q < IH � r

IS + q < IH q < IH � IS

The size of the payout q determines the fullness of the insurance contract. A contract with a payout that fully covers the spread between IH and IS is full, while contracts with payouts that do not fully cover this difference are partial. The closer an insurance contract’s payout q comes to equaling IH � IS, the fuller we say that contract is.

Just as we think of the fairness of a contract as its effective price, we can think of the fullness of a contract as its effective quantity. Fuller contracts offer higher quantities of insurance, in the sense that they provide greater income certainty and produce greater expected utility.

Figure 7.4 compares an individual’s income and utility under three different insurance contracts that are all actuarially fair but vary in their degree of fullness:

• No insurance: the individual receives either IS or IH , and has expected utility at A.

• Partial insurance: the individual receives either I PS or I P H , and has expected

utility at AP.