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INSURANCE. Insurance Reduces Risk. What is risk? Something to do with variability of returns? Does insurance have variable returns? Insurance’s variability reduces overall portfolio variability. The Expected Value Concept. Actuarial Fairness.

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insurance reduces risk
Insurance Reduces Risk
  • What is risk?
    • Something to do with variability of returns?
    • Does insurance have variable returns?
    • Insurance’s variability reduces overall portfolio variability
actuarial fairness
Actuarial Fairness
  • Game/insurance with fee/premium equal to expected value of outcomes
risk preferences
Risk Preferences
  • A RISK NEUTRAL person would pay as much as $.50 for a coin toss paying $1 heads, $0 tails
  • A RISK LOVING person would pay more than $.50
  • A RISK AVERSE person would pay, at most, less than $.50
risk aversion and declining marginal utility
Risk Aversion and Declining Marginal Utility
  • Gain from winning a dollar less than loss from losing a dollar
a risk averse utility function
A Risk Averse Utility Function

Total

Utility

200

140

Utility

20

10

0

Wealth ($000)

total and marginal utility
Total and Marginal Utility

TU

Utility

Wealth

0

Utility

MU

0

Wealth

certainty equivalent of expected value
Certainty Equivalent of Expected Value
  • Utility if wealth actually equaled expected value of wealth
an example
An Example
  • Wealth = $20,000 if well with probability .95, $10,000 if ill with probability .05
  • EU = .95 X (utility of $20,000) + .05 x (utility of $10,000)
  • EU = (.95 x 200) + (.05 x 140)
  • EU = 190 + 7 = 197
  • EV = (.95 x $20,000) + (.05 x $10,000)
  • EV = $19,000 + $500 = $19,500
  • CE = 199
the graph of the example
The Graph of the Example

B

D

200

199

FE shows max

willingness to

pay for insurance

197

E

C

F

A

Utility

140

0

17

19.5

10

20

Wealth ($000)

buying insurance
Buying Insurance
  • Suppose our consumer is offered the opportunity to insure against this loss for $500
    • Paying the premium means income will be $19,500 ($20,000 minus the $500 premium) no matter what happens
  • Utility at $19,500 (point D) exceeds expected utility (point C)
    • utility of 199 versus 197
  • Willing to pay up to $3,000
    • Distance FE
    • Any amount less than $3,000 gives more utility than the expected utility of $20,00 with probability .95 and $10,000 with probability .05
the demand for insurance
The Demand for Insurance
  • How much insurance will an individual buy?
  • Notation:
    • p = the probability of illness
    • W = initial wealth
    • L = financial loss because of illness
  • Without insurance:
    • EU = p x (utility of net wealth ill) + (1-p) x (utility of net wealth well)
    • EU = p U(W - L) + (1-p) U(W)
slide15
With insurance:

Wealth ill = Wealth (W) - Loss (L) - insurance premium (q) + payment from insurance (q), where  is the premium rate and q is the coverage

slide16

Wealth ill = W - L - q + q

  • = W - L - (1 - )q
  • Wealth well = Wealth (W) - premium (q)
  • Wealth well = W - q
  • Thus, Expected Utility = p U(W - L + (1 - )q) [1] + (1 - p) U(W - q) [2]
  • Expression [1] is related to the benefit of insurance and [2] is related to the cost
  • Individual will buy coverage (q) where MB = MC
marginal benefit and marginal cost
Marginal Benefit and Marginal Cost
  • MB related to expression [1]
  • As coverage (q) increases, the marginal utility of the extra money falls
  • MC related to expression [2]
  • As q increases, wealth when well decreases, so utility forgone rises
graph of mb and mc
Graph of MB and MC

Utils

Marginal cost (in utils)

MC

A

Marginal benefit (in utils)

MB

0

q*

Coverage purchased

what happens if premium rate rises
What Happens if Premium Rate () Rises?
  • MB shifts down to MB1
  • MC shifts up to MC1
  • Equilibrium moves from A to B
  • Less coverage purchased
premium rises
Premium Rises

Utils

Marginal cost (in utils)

MC

MC1

B

A

MB

MB1

0

q1

q2

Coverage purchased

what happens if expected loss l increases
What Happens if Expected Loss (L) Increases?
  • MB shifts up to MB2 because at lower wealth, MU of any additional q is greater
  • L not in cost expression so MC does not change
  • Equilibrium moves from A to C
  • More coverage purchased
expected loss increases
Expected Loss Increases

Utils

Marginal cost (in utils)

MC

C

A

MB2

MB

0

q2

q1

Coverage purchased

what happens if wealth w increases
What happens if Wealth (W) Increases?
  • At any level of coverage (q), both the marginal utility of q when ill (MB) and the marginal utility of wealth forgone when well (MC) fall
  • Equilibrium moves from A to D
  • Affect on coverage purchased ambiguous
    • coverage goes up in following graph
    • would have gone down if fall in MB were larger or fall in MC smaller
wealth increases
Wealth Increases

Utils

Marginal cost (in utils)

MC1

MC2

A

MB1

MB2

D

0

q1

q2

Coverage purchased

the supply of insurance
The Supply of Insurance
  • What determines the premium rate ()?
  • As a point of departure, assume perfect competition
    • in long run, perfectly competitive firms earn zero profit
    • what  results in zero profit?
  • If representative customer is well, insurer earns q dollars
slide26

If customer gets ill, insurer loses q - q, or (1 - )q dollars

  • Either way, insurer incurs loading cost t
    • cost of servicing transactions
  • Exp profit = (1 – p)q – p(1 - )q - t
slide27

Assume perfect competition (zero profit)

  • Then (1 – p)q – p(1 - )q - t = 0
  • Or q - pq – pq + pq - t = 0
  • Or q - pq - t = 0
  • Or  = p + t/q
  • Thus, premium rate () equals the probability of illness plus loading cost as a proportion of coverage
slide28

E.g., if the probability of illness is .05 and loading costs are 10% of coverage, then the premium will be $.15 for every dollar of coverage

  • If more is charged, other insurers will take all the business
  • if less is charged, profits will be lost
  • If loading costs are zero, insurance will be actuarially fair:  = p
optimal coverage q
Optimal Coverage (q)
  • What amount of insurance (q) will consumer choose?
  • Recall that

EU = pU(W – L + (1 - )q)+ (1 – p)U(W - q)

  • To find max EU, take derivative of EU with respect to q and set equal to zero:

p(1 - )MU(W – L + (1 - )q) – (1 – p)MU(W - q) = 0

where MU(. . .) refers to marginal utility (i.e., derivative of U)

slide30
MU(W – L + (1 - )q) = XMU(W - q)

where X = [(1 – p)]/[p(1 - )]

slide31

If  = p (actuarially fair), then X = 1 and MU(W - L + (1 - )q) = MU(W - q)

  • This can only happen if W - L + (1 - )q = W - q or q = L
  • So if  = p, the consumer will fully insure
slide32

Recall, though, that under perfect competition, insurers will set  = p + t/q

  • So if there are loading costs, consumers will not fully insure
  • To see exactly why, return to the MB = MC relationship
slide33

p(1 - )MU(W - L + (1 - )q) = (1 - p)MU(W - q)

  • Recall that under perfect competition (1 - p) = p(1 - ) + t/q
  • Substitute this expression for (1 - p) into equation above to get
  • p(1 - )MU(W - L + (1 - )q) = [p(1 - ) + t/q]MU(W - q)
slide34
divide through by p(1 - ) to get

MU(W – L + (1 - )q) =

[p(1 - ) + t/q]/[p(1 - )]MU(W - q)

Now [p(1 - ) + t/q]/[p(1 - )] = 1 + [t/q]/[p(1 - )]

= 1 + t/[qp(1 - )]

So,

MU(W – L + (1 - )q) = (1 + Z)MU(W - q)

where Z = t/[qp(1 - )]

slide35

If loading costs (t) are zero, then consumer fully insures (q = L)

  • If t > 0, then Z > 0 and MC shifted up by (1 + Z)
  • Thus, q < L (i.e., the consumer underinsures)
phew here s the bottom line
PHEW!!! Here’s the Bottom Line
  • If loading costs (t) are zero, perfect competition forces insurers to charge actuarially fair premiums
  • If premiums are actuarially fair, consumers will fully insure
  • If there are loading costs, premiums will be higher and consumers will buy less than full insurance
moral hazard
Moral Hazard
  • Analysis assumes, so far, that the loss (L) is fixed
  • What if L is not fixed?
  • Say L is affected by the health care price faced by consumer?
illustration of moral hazard
Illustration of Moral Hazard

Inelastic Demand

Price-Sensitive Demand

$

$

D

D

P1

P1

Health care

Health care

Q1

Q1

Q2

slide39

If insurer charges premium based on L = P1Q1, it will lose money because loss will actually be P1Q2

  • If insurer charges premium based on L = P1Q2, consumer may not buy since premium may exceed what he would pay for health care in absence of insurance
testable hypotheses
Testable Hypotheses
  • There will be more complete coverage the less elastic is demand
  • Insurance develops first for those services that are less elastic
  • Cross sectional data support first hypothesis
  • Time-series data support second
the effect of deductibles
The Effect of Deductibles
  • No effect if deductible is small, allowing consumer to buy health care at zero marginal price once deductible is paid
  • Causes consumer to self-insure if deductible is so large that the gain from being able to buy at zero marginal price less than deductible
coinsurance
Coinsurance
  • Insurance requiring consumer to pay a percentage of the loss
  • E.g., a 20% coinsurance requires consumer to pay 20% of the cost of his consumption of health care
  • What does coinsurance do to demand?
illustration of coinsurance
Illustration of Coinsurance

D (100% coinsurance)

$

D (< 100% coinsurance)

A

B

P1

S

P2

C

0

Health Care

Q1

Q2

slide44

Insurance causes demand to swivel out causing more health care to be demanded

  • The lower is coinsurance, the less elastic is demand
  • At 0% coinsurance, demand becomes totally inelastic
welfare loss
Welfare Loss
  • If no insurance, demand reflects all benefits (assuming no externalities)
  • Insurance causes welfare loss because market demand does not reflect benefits of health care
illustration of welfare loss
Illustration of Welfare Loss

$

D (100%)

D (20%)

S

Deadweight Loss

0

Q1

Q2

Health Care

slide47

Thus, insurance results in over-allocation to insured forms of health care at expense of non-insured forms (good nutrition, exercise) and also at expense of non-health care goods and services