INSURANCE

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INSURANCE
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Insurance Reduces Risk

• What is risk?
• Something to do with variability of returns?
• Does insurance have variable returns?
• Insurances variability reduces overall portfolio
  variability

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The Expected Value Concept
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Actuarial Fairness

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

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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

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Risk Aversion and Declining Marginal Utility

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

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A Risk Averse Utility Function
Total Utility
200
140
Utility
20
10
0
Wealth (000)
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Total and Marginal Utility
TU
Utility
Wealth
0
Utility
MU
Wealth
0
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Expected Utility of Wealth
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Certainty Equivalent of Expected Value

• Utility if wealth actually equaled expected value
  of wealth

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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

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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
19.5
17
20
10
Wealth (000)
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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

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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)

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• 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

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• 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

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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

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Graph of MB and MC
Utils
Marginal cost (in utils) MC
A
Marginal benefit (in utils) MB
0
q
Coverage purchased
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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

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Premium Rises
Utils
Marginal cost (in utils) MC
MC1
A
B
MB
MB1
0
q1
q2
Coverage purchased
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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

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Expected Loss Increases
Utils
Marginal cost (in utils) MC
C
A
MB2
MB
0
q2
q1
Coverage purchased
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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

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Wealth Increases
Utils
Marginal cost (in utils) MC1
MC2
A
MB1
MB2
D
0
q1
q2
Coverage purchased
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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

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• 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

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• 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

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• 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

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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)

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• MU(W L (1 - ?)q) XMU(W - ?q)
• where X (1 p)?/p(1 - ?)

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• 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

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• 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

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• 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/qMU(W - ?q)

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• 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 - ?)

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• If loading costs (t) are zero, then consumer
  fully insures (q L)
• If t gt 0, then Z gt 0 and MC shifted up by (1 Z)
• Thus, q lt L (i.e., the consumer underinsures)

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PHEW!!! Heres 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

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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?

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Illustration of Moral Hazard
Inelastic Demand
Price-Sensitive Demand


D
D
P1
P1
Health care
Health care
Q1
Q2
Q1
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• 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

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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

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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

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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?

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Illustration of Coinsurance
D (100 coinsurance)

D (lt 100 coinsurance)
A
B
P1
S
P2
C
0
Health Care
Q1
Q2
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• 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

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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

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Illustration of Welfare Loss

D (100)
D (20)
S
Deadweight Loss
0
Q1
Q2
Health Care
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• 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