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CHAPTER 7

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 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.

Income and utility



Graphically,

 Utility increasing with income U’(I) > 0  Marginal utility decreasing U’’(I) < 0

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  (^) I S = income if she does get sick  (^) I H > IS = income if she remains healthy

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 $ 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) = $ E[B] = 1(250) = $

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.

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.

Expected utility without

insurance



Lottery scenario similar to case of insurance

customer

 (^) She gains a high income I H if healthy, and low income I S 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(I S ) + (1- p) U(I H

What if p lies between 0 and



For p between 0 and 1, expected utility falls

on a line segment between S and H

Ex: p = 0.

 For p = 0.25, person’s expected income is: E[I] = 0.25·I S

+ (1 - .25)·I

H  Utility at that expected income is E[U(I)] (Point A)

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

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: I S

• q – r  (^) Healthy: I H
• 0 – r

Full insurance



Full insurance means no income uncertainty

I

S

’ = I

H



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)

Full insurance payout



State independence implies

I

H

’ = I

S



So

I

H

+ 0 – r = I

S

+ q – r

I

H

= I

S

+ q

q = I

H

• I S 

The payout from a full insurance contract is

difference between incomes without insurance