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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

- Game/insurance with fee/premium equal to expected value of outcomes

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

- Gain from winning a dollar less than loss from losing a dollar

Certainty Equivalent of Expected Value

- Utility if wealth actually equaled expected value of wealth

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

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

- 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

- 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)

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

- = 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

- 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

Utils

Marginal cost (in utils)

MC

A

Marginal benefit (in utils)

MB

0

q*

Coverage purchased

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

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

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

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

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

Assume perfect competition (zero profit)

- Then (1 – p)q – p(1 - )q - t = 0
- Or q - pq – pq + pq - 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

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)

- 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)

MU(W – L + (1 - )q) = XMU(W - q)

where X = [(1 – p)]/[p(1 - )]

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

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

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)

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 - )]

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

- 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

- 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

Inelastic Demand

Price-Sensitive Demand

$

$

D

D

P1

P1

Health care

Health care

Q1

Q1

Q2

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

- 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

- 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

- 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?

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

- If no insurance, demand reflects all benefits (assuming no externalities)
- Insurance causes welfare loss because market demand does not reflect benefits of health care

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

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